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Obtaining the quadratic shown below earns the first method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(x^2+7x+12)(x+2)$"},{"latex":"$$x^3+7x^2+12x+2x^2+14x+24$"}]},"order":1,"title":"Multiply the third bracket","description":"Now multiply every term in the quadratic by both terms in the third bracket. Showing this full expansion earns the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x^3+7x^2+12x+2x^2+14x+24$"},{"latex":"$$x^3+9x^2+26x+24$"}]},"order":2,"title":"Collect like terms","description":"Collect the like terms in the expanded expression. The simplified result earns the accuracy mark (A1)."}]}],"generatedAt":"2026-07-20T11:08:49.979Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b03f8322-05bc-41db-accc-85c67f6ab4aa:sequential_expansion","label":"Sequential Bracket Expansion","steps":[{"type":"text","order":0,"title":"Choose two brackets","content":"To expand triple brackets using sequential bracket expansion, we first choose any two of the brackets to expand. Let's choose the first two:\n\n$$(x + 3)(x + 4)$$","highlights":[{"text":"$$(x + 3)(x + 4)(x + 2)$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first-two}{\\color{@sky}{(x + 3)(x + 4)}} = \\htmlData{anchor=term-1}{x^2} + \\htmlData{anchor=term-2}{4x} + \\htmlData{anchor=term-3}{3x} + \\htmlData{anchor=term-4}{12}$"},{"latex":"$$\\phantom{(x + 3)(x + 4)} = \\htmlData{anchor=simplified}{\\color{@sky}{x^2 + 7x + 12}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"first-two"},{"color":"sky","style":"box","anchor":"simplified"}],"connections":[{"to":"simplified","from":"first-two","color":"sky","label":"expanded","style":"solid"}]},"order":1,"title":"Expand the first two","description":"Expand $(x + 3)(x + 4)$ by multiplying each term in the first bracket by each term in the second bracket, and then simplify by collecting the like $x$ terms ($4x + 3x = 7x$). This earns the first method mark (M1)."},{"type":"text","order":2,"title":"Multiply by third bracket","content":"Now, we multiply this resulting quadratic expression by the remaining linear bracket, $(x + 2)$:\n\n$$(x^2 + 7x + 12)(x + 2)$$\n\nTo do this, we multiply each of the three terms in the quadratic by both terms in the linear bracket. Let's do this step-by-step:\n\n- Multiply by $x$:\n$$x(x^2 + 7x + 12) = x^3 + 7x^2 + 12x$$\n\n- Multiply by $2$:\n$$2(x^2 + 7x + 12) = 2x^2 + 14x + 24$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(\\color{@sky}{x^3} + \\color{@amber}{7x^2} + \\color{@purple}{12x}) + (\\color{@amber}{2x^2} + \\color{@purple}{14x} + \\color{@teal}{24})$"},{"latex":"$$= \\htmlData{anchor=x3}{\\color{@sky}{x^3}} + \\htmlData{anchor=x2}{\\color{@amber}{9x^2}} + \\htmlData{anchor=x1}{\\color{@purple}{26x}} + \\htmlData{anchor=const}{\\color{@teal}{24}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x3"},{"color":"amber","style":"text-color","anchor":"x2"},{"color":"purple","style":"text-color","anchor":"x1"},{"color":"teal","style":"text-color","anchor":"const"}],"connections":[]},"order":3,"title":"Combine and simplify","description":"Combine the results together (M1) and collect the like terms:\n- For $x^2$: $7x^2 + 2x^2 = 9x^2$\n- For $x$: $12x + 14x = 26x$\n\nThis gives us our final simplified expression (A1)."}]},{"id":"b03f8322-05bc-41db-accc-85c67f6ab4aa:grid_method","label":"Grid Method","steps":[{"type":"text","order":0,"title":"Multiply the first two brackets","content":"To expand $(x + 3)(x + 4)(x + 2)$, we will start by expanding the first two brackets: \n$$(x + 3)(x + 4)$$\n\nWe set up a $2 \\times 2$ grid with $(x + 3)$ along the top and $(x + 4)$ down the side:\n\n| $\\times$ | $x$ | $+3$ |\n| :---: | :---: | :---: |\n| **$x$** | $x^2$ | $+3x$ |\n| **$+4$** | $+4x$ | $+12$ |\n\nAdding the terms inside the grid gives us:\n$$x^2 + 3x + 4x + 12$$","highlights":[{"text":"(x + 3)(x + 4)(x + 2)","location":"specification"}]},{"type":"text","order":1,"title":"Simplify the quadratic expression","content":"Now we simplify the expression by combining the like terms ($3x$ and $4x$):\n\n$$3x + 4x = 7x$$\n\nSo, our simplified quadratic expression is:\n$$x^2 + 7x + 12$$\n\n*(This earns the first mark M1)*"},{"type":"text","order":2,"title":"Set up the second grid","content":"Next, we multiply this resulting quadratic by the remaining linear bracket, $(x + 2)$:\n$$(x^2 + 7x + 12)(x + 2)$$\n\nWe set up a $2 \\times 3$ grid with the quadratic expression $(x^2 + 7x + 12)$ along the top and $(x + 2)$ down the side:\n\n| $\\times$ | $x^2$ | $+7x$ | $+12$ |\n| :---: | :---: | :---: | :---: |\n| **$x$** | $x^3$ | $+7x^2$ | $+12x$ |\n| **$+2$** | $+2x^2$ | $+14x$ | $+24$ |\n\n*(Setting up this multiplication earns the second mark M1)*"},{"type":"text","order":3,"title":"Simplify to get the final answer","content":"Finally, we collect all the terms from inside our second grid:\n$$x^3 + 7x^2 + 2x^2 + 12x + 14x + 24$$\n\nNow we combine the like terms:\n- Combine $x^2$ terms: $7x^2 + 2x^2 = 9x^2$\n- Combine $x$ terms: $12x + 14x = 26x$\n\nThis gives us our fully simplified cubic expression:\n$$x^3 + 9x^2 + 26x + 24$$\n\n*(This earns the final accuracy mark A1)*"}]},{"id":"b03f8322-05bc-41db-accc-85c67f6ab4aa:polynomial_identity","label":"Using Polynomial Identity","steps":[{"type":"text","order":0,"title":"Identify the identity","content":"To expand and simplify the expression $(x + 3)(x + 4)(x + 2)$, we can use the polynomial identity for the product of three binomials:\n\n$$(x + a)(x + b)(x + c) = x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$$","highlights":[{"text":"(x + 3)(x + 4)(x + 2)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$a = \\htmlData{anchor=val-a}{\\color{@sky}{3}}, \\quad b = \\htmlData{anchor=val-b}{\\color{@amber}{4}}, \\quad c = \\htmlData{anchor=val-c}{\\color{@purple}{2}}$"},{"latex":"$$(x + \\htmlData{anchor=var-a}{\\color{@sky}{a}})(x + \\htmlData{anchor=var-b}{\\color{@amber}{b}})(x + \\htmlData{anchor=var-c}{\\color{@purple}{c}}) = x^3 + (\\htmlData{anchor=sum-a}{\\color{@sky}{a}} + \\htmlData{anchor=sum-b}{\\color{@amber}{b}} + \\htmlData{anchor=sum-c}{\\color{@purple}{c}})x^2 + (\\htmlData{anchor=prod-ab}{\\color{@sky}{a}}\\htmlData{anchor=prod-ba}{\\color{@amber}{b}} + \\htmlData{anchor=prod-bc}{\\color{@amber}{b}}\\htmlData{anchor=prod-cb}{\\color{@purple}{c}} + \\htmlData{anchor=prod-ca}{\\color{@purple}{c}}\\htmlData{anchor=prod-ac}{\\color{@sky}{a}})x + \\htmlData{anchor=term-abc}{\\color{@sky}{a}}\\htmlData{anchor=term-bca}{\\color{@amber}{b}}\\htmlData{anchor=term-cab}{\\color{@purple}{c}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val-a"},{"color":"amber","style":"text-color","anchor":"val-b"},{"color":"purple","style":"text-color","anchor":"val-c"}],"connections":[{"to":"var-a","from":"val-a","color":"sky","label":"a = 3","style":"solid"},{"to":"var-b","from":"val-b","color":"amber","label":"b = 4","style":"solid"},{"to":"var-c","from":"val-c","color":"purple","label":"c = 2","style":"solid"}]},"order":1,"title":"Substitute the values","description":"We identify $a = 3$, $b = 4$, and $c = 2$ from our given expression $(x + 3)(x + 4)(x + 2)$ and substitute them into the identity."},{"type":"text","order":2,"title":"Calculate the coefficients","content":"Now, we compute each coefficient term:\n\n1. **Coefficient of $x^2$**:\n $$a + b + c = 3 + 4 + 2 = 9$$\n\n2. **Coefficient of $x$**:\n $$ab + bc + ca = (3 \\times 4) + (4 \\times 2) + (2 \\times 3) = 12 + 8 + 6 = 26$$\n\n3. **Constant term**:\n $$abc = 3 \\times 4 \\times 2 = 24$$"},{"type":"text","order":3,"title":"Write the final expression","content":"Plugging these values back into the polynomial identity, we get:\n\n$$x^3 + 9x^2 + 26x + 24$$\n\nThis is the fully expanded and simplified form of the expression."}]}],"generatedAt":"2026-06-13T03:50:50.164Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":84},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/c3d3ce26-5893-4a56-b88e-f55e9ce017fc.mp4","durationSeconds":83,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/c3d3ce26-5893-4a56-b88e-f55e9ce017fc.jpg"}},{"id":"cb807ffe-aab1-4210-9799-0e7e3ae64585","revisionId":"fc387abf-e2b8-409a-9182-716c29524a33","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"2a5cec5c-2d37-4e63-86db-d56c0ecaa2b9","label":"a","content":"Expand and Simplify $(x + 3)(x + 5)(x + 2)$","markscheme":"- Expands two brackets to get $x^2 + 8x + 15$ or $x^2 + 5x + 6$ or $x^2 + 7x + 10$ M1\n- Multiplies the resulting quadratic by the third linear bracket M1\n- Final simplified expression $x^3 + 10x^2 + 31x + 30$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Attempts to expand two of the brackets to obtain a quadratic expression with at least two correct terms.","code":"M","acceptable":["$$x^2 + 8x + 15$","$$x^2 + 5x + 6$","$$x^2 + 7x + 10$"]},{"id":"p2","marks":1,"description":"Multiplies their quadratic expression by the third linear bracket, resulting in at least 4 correct terms in an unsimplified cubic expression.","code":"M"},{"id":"p3","marks":1,"description":"Provides the fully simplified expression $x^3 + 10x^2 + 31x + 30$.","code":"A"}],"finalAnswers":["x^3 + 10x^2 + 31x + 30"],"generatedAt":"2026-05-06T14:00:49.592Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":85},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/cb807ffe-aab1-4210-9799-0e7e3ae64585.mp4","durationSeconds":96,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/cb807ffe-aab1-4210-9799-0e7e3ae64585.jpg"}}],"subjectCode":"gcse_maths","summaries":[{"id":"9aad1f11-5bb3-4968-9a92-67ab6ecda6c1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1b52c5a3-f72e-4a10-a582-3f4f2f6ee2e3","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6b60c54b-a998-4287-a416-8a95faabec89","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6ca6baa2-9041-46a7-b4e4-34413367d1ca","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6ffb412c-e172-48a7-a8f1-cefa7fb60423","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"74dd7fe6-5692-4f20-a6cc-99363d6ae86f","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"7500b37d-c446-49e6-ac2a-c03abdac7d18","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a9e35d4f-e262-4e1a-b2b3-ebfa91a89395","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c3d3ce26-5893-4a56-b88e-f55e9ce017fc","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c5d3e484-fb33-4c57-aae1-97d9f7a71ebe","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"cb807ffe-aab1-4210-9799-0e7e3ae64585","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d52999bd-b641-487f-8f42-9b4b8cd63cee","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f5b256be-46cb-4f45-ae2e-1537270ad749","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f7d8c55d-6bbb-4b7c-b431-7c9701d43c65","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"f7f78c26-ea4e-4864-8d79-b103176ad6b7","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b9b45640-efe2-46d9-869e-d48ed30b79d4","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"24352beb-b4e4-4f9c-80e4-591f4f7bd69a","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3850b855-d469-43c5-98e8-c065ed453955","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3a34c51d-8cb1-47a3-ab02-dc3f9fbd7979","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4538684e-d85f-4688-97c9-483dde3b18b2","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"48faf46f-8872-4cce-927c-8b69b6a6b53d","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4db43ee6-ed03-4dd6-882f-3e7617a8320b","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4e320c7f-c86e-4898-87d8-edcef6307b1d","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"652e534e-f2a2-48d5-aacc-67e472a90a26","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"80f5e67b-c747-4ec1-9250-bae342ae8cde","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9bb1a3f1-e05d-47dd-bf6c-e9be2697680c","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bb5652d1-757f-4ae5-8441-61ce99107ff8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d119def4-4009-4f43-9149-a2640ebdeb3c","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d4a3260b-ca2a-4ecb-ae7c-0cdc6305de86","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"dd0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Triple Brackets","topicId":"fc35be10-1d50-43e5-b80e-9d81f4c1386f","topicSidebar":"$L3a","topicSlug":"grade-6-6-5-expanding-triple-brackets","topicTitle":"Expanding Triple Brackets"}]}],"$L3b","$L3c"]}]

Expanding Triple Brackets

Question 1

Expanding Triple Brackets Questions

Expanding Triple Brackets

Question 1

Expanding Triple Brackets Questions