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Brackets","calculatorUse":null,"isCalculation":true,"parts":[{"id":"4876ee60-3045-41b4-b5e7-3d9918f00848","label":"0","content":"Expand and Simplify $(3x + 1)(x + 2)(x - 4)$","markscheme":"- Expands two brackets correctly, e.g., $(x^2 - 2x - 8)$ or $(3x^2 + 7x + 2)$ M1\n- Multiplies by the third bracket M1\n- Final simplified expression $3x^3 - 5x^2 - 26x - 8$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly expands any two brackets to a quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the quadratic by the remaining linear bracket.","code":"M"},{"id":"p3","marks":1,"description":"Final simplified answer $3x^3 - 5x^2 - 26x - 8$.","code":"A"}],"finalAnswers":["3x^3 - 5x^2 - 26x - 8"],"generatedAt":"2026-05-07T02:01:01.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"successive-expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Expand the first pair","content":"Start with the last two brackets. Multiply each term in $(x+2)$ by each term in $(x-4)$:\n\n$$(x+2)(x-4)=x^2-4x+2x-8=x^2-2x-8$$\n\nSo the original product becomes\n\n$$(3x+1)(x+2)(x-4)=(3x+1)(x^2-2x-8).$$\n\nThis correct expansion of two brackets earns the first method mark (M1).","highlights":[{"text":"(x + 2)(x - 4)","location":"specification"}]},{"type":"text","order":1,"title":"Multiply and collect","content":"Multiply the quadratic by both terms in $(3x+1)$, then collect like terms:\n\n$$(3x+1)(x^2-2x-8)=3x^3-6x^2-24x+x^2-2x-8$$\n\n$$=3x^3-5x^2-26x-8$$\n\nThis is multiplication by the third bracket, earning the second method mark (M1)."},{"type":"text","order":2,"title":"State the final answer","content":"Therefore,\n\n$$\\boxed{3x^3-5x^2-26x-8}$$\n\nThis fully simplified expression earns the final accuracy mark (A1)."}]}],"generatedAt":"2026-09-01T09:17:43.436Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4876ee60-3045-41b4-b5e7-3d9918f00848:successive-expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Expand the first pair","content":"Start with the last two brackets. Multiply each term in $(x+2)$ by each term in $(x-4)$:\n\n$$(x+2)(x-4)=x^2-4x+2x-8=x^2-2x-8$$\n\nSo the original product becomes\n\n$$(3x+1)(x+2)(x-4)=(3x+1)(x^2-2x-8).$$\n\nThis correct expansion of two brackets earns the first method mark (M1).","highlights":[{"text":"(x + 2)(x - 4)","location":"specification"}]},{"type":"text","order":1,"title":"Multiply and collect","content":"Multiply the quadratic by both terms in $(3x+1)$, then collect like terms:\n\n$$(3x+1)(x^2-2x-8)=3x^3-6x^2-24x+x^2-2x-8$$\n\n$$=3x^3-5x^2-26x-8$$\n\nThis is multiplication by the third bracket, earning the second method mark (M1)."},{"type":"text","order":2,"title":"State the final answer","content":"Therefore,\n\n$$\\boxed{3x^3-5x^2-26x-8}$$\n\nThis fully simplified expression earns the final accuracy mark (A1)."}]}],"generatedAt":"2026-09-01T09:17:46.686Z","modelVersion":"gpt-5.6-sol+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ca392bc2-3d0b-4ba7-b0d5-81d9b5790693","revisionId":"fcad60c6-b71f-416f-a179-ee92f3f0d030","questionNumber":"11","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","tier":null,"totalMarks":3,"sourceTitle":"Expanding Triple Brackets","calculatorUse":null,"isCalculation":true,"parts":[{"id":"21b6d1d7-6b61-4234-bdbc-2a927e3748b7","label":"0","content":"Show that $(2x + 3)(5x + 2)(x - 5) = 10x^3 - 31x^2 - 89x - 30$ for all values of $x$.","markscheme":"- Expands two brackets correctly, e.g., $(5x^2 - 23x - 10)$ M1\n- Multiplies by $(2x + 3)$ with at least 4 terms correct M1\n- Completes expansion to show the required expression $10x^3 - 31x^2 - 89x - 30$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correctly expands any two brackets to a quadratic.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the quadratic by the remaining linear bracket.","code":"M"},{"id":"p3","marks":1,"description":"Fully simplifies to reach the target expression $10x^3 - 31x^2 - 89x - 30$.","code":"A"}],"finalAnswers":["10x^3 - 31x^2 - 89x - 30"],"generatedAt":"2026-05-07T02:01:01.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"successive_expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Choose two brackets","content":"Start with the two brackets on the right. Expanding them first leaves one linear bracket to multiply at the end:\n\n$$(5x+2)(x-5)$$\n\nUse distribution: multiply **every term** in the first bracket by every term in the second. Getting this quadratic earns the first method mark (M1).","highlights":[{"text":"$$(2x + 3)(5x + 2)(x - 5)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(5x+2)(x-5)=5x(x-5)+2(x-5)"},{"latex":"=5x^2-25x+2x-10"},{"latex":"(5x+2)(x-5)=5x^2+\\htmlData{anchor=combined-x}{(-23x)}-10"}],"highlights":[{"color":"amber","style":"box","anchor":"combined-x"}]},"order":1,"title":"Expand the first pair","description":"Multiply each term in the first bracket by the whole second bracket, then collect the two terms containing $x$."},{"type":"text","order":2,"title":"Multiply the remaining bracket","content":"Now multiply **both** $2x$ and $3$ by all three terms of the quadratic. Write all six products before collecting like terms; this is the safest way not to miss a term.\n\nThis full multiplication is the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(2x+3)(5x^2-23x-10)=10x^3-46x^2-20x+15x^2-69x-30"},{"latex":"=10x^3+\\htmlData{anchor=square-terms}{(-46x^2+15x^2)}+\\htmlData{anchor=x-terms}{(-20x-69x)}-30"},{"latex":"=10x^3-31x^2-89x-30"}],"highlights":[{"color":"sky","style":"box","anchor":"square-terms"},{"color":"amber","style":"box","anchor":"x-terms"}]},"order":3,"title":"Collect like terms","description":"Group terms with the same power of $x$: cubic terms, square terms, $x$ terms, then constants."},{"type":"text","order":4,"title":"State the identity","content":"The expansion is exactly the expression required:\n\n$$\\boxed{(2x+3)(5x+2)(x-5)=10x^3-31x^2-89x-30}$$\n\nBecause the brackets were expanded without choosing a particular value of $x$, this holds for **all** values of $x$. Matching the required cubic earns the accuracy mark (A1)."}]}],"generatedAt":"2026-09-01T09:09:17.619Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"21b6d1d7-6b61-4234-bdbc-2a927e3748b7:successive_expansion","label":"Expand brackets successively","steps":[{"type":"text","order":0,"title":"Choose two brackets","content":"Start with the two brackets on the right. Expanding them first leaves one linear bracket to multiply at the end:\n\n$$(5x+2)(x-5)$$\n\nUse distribution: multiply **every term** in the first bracket by every term in the second. Getting this quadratic earns the first method mark (M1).","highlights":[{"text":"$$(2x + 3)(5x + 2)(x - 5)","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"(5x+2)(x-5)=5x(x-5)+2(x-5)"},{"latex":"=5x^2-25x+2x-10"},{"latex":"(5x+2)(x-5)=5x^2+\\htmlData{anchor=combined-x}{(-23x)}-10"}],"highlights":[{"color":"amber","style":"box","anchor":"combined-x"}]},"order":1,"title":"Expand the first pair","description":"Multiply each term in the first bracket by the whole second bracket, then collect the two terms containing $x$."},{"type":"text","order":2,"title":"Multiply the remaining bracket","content":"Now multiply **both** $2x$ and $3$ by all three terms of the quadratic. Write all six products before collecting like terms; this is the safest way not to miss a term.\n\nThis full multiplication is the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"(2x+3)(5x^2-23x-10)=10x^3-46x^2-20x+15x^2-69x-30"},{"latex":"=10x^3+\\htmlData{anchor=square-terms}{(-46x^2+15x^2)}+\\htmlData{anchor=x-terms}{(-20x-69x)}-30"},{"latex":"=10x^3-31x^2-89x-30"}],"highlights":[{"color":"sky","style":"box","anchor":"square-terms"},{"color":"amber","style":"box","anchor":"x-terms"}]},"order":3,"title":"Collect like terms","description":"Group terms with the same power of $x$: cubic terms, square terms, $x$ terms, then constants."},{"type":"text","order":4,"title":"State the identity","content":"The expansion is exactly the expression required:\n\n$$\\boxed{(2x+3)(5x+2)(x-5)=10x^3-31x^2-89x-30}$$\n\nBecause the brackets were expanded without choosing a particular value of $x$, this holds for **all** values of $x$. Matching the required cubic earns the accuracy mark (A1)."}]}],"generatedAt":"2026-09-01T09:09:17.972Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"64df4672-97ef-49d3-b8f4-76fec52a2de0","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4bc64585-6eb9-4a8f-b764-ab56a3d6130e","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"78f75685-fcb2-4bff-a809-54d392f9efea","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"354889bd-222b-4947-b54b-528370558cd9","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0fd7b290-efd6-4943-96c7-0856f8590076","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8fe72371-59f3-44b6-9690-13451b80b444","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bb22e8bb-f94a-4614-a71f-aeabd147d1d5","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"43fccfe1-09e4-4c0c-91d8-d4a6d3232863","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2fdef4b0-1217-4b8d-aba4-596aa5cc5d97","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88d03f80-e249-4b37-a6f7-b0e553166a40","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ca392bc2-3d0b-4ba7-b0d5-81d9b5790693","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9f293ed4-1f33-4831-83e2-f7ac1340982c","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"361fff12-8aa1-4e7c-8bb6-1f071bed0bcc","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4d6510f2-d4bb-4bce-bfd8-290ae3662f18","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true}],"title":"Expanding Triple Brackets","topicId":"ab12d12d-260e-4889-9123-0582ebb0c140","topicSidebar":"$L3c","topicSlug":"grade-6-6-4-expanding-triple-brackets","topicTitle":"Expanding Triple Brackets"}]}],"$L3d","$L3e"]}]