ighlights":[{"color":"sky","style":"underline","anchor":"pair"},{"color":"sky","style":"box","anchor":"quadratic"}],"connections":[{"to":"quadratic","from":"pair","color":"sky","label":"expanded pair","style":"solid"}]},"order":1,"title":"Multiply the linear pair","description":"We expand the pair $(y - 3)(y + 2)$ by multiplying each term in the first bracket by each term in the second bracket and then combining the like terms. This earns the first mark M1."},{"type":"text","order":2,"title":"Multiply by the last bracket","content":"Now we substitute our quadratic expression back and multiply it by the remaining bracket $(y + 2)$:\n\n$$\\begin{aligned}\n(y^2 - y - 6)(y + 2) &= y(y^2 - y - 6) + 2(y^2 - y - 6) \\\\\n&= y^3 - y^2 - 6y + 2y^2 - 2y - 12\n\\end{aligned}$$\n\nThis full expansion of the remaining brackets earns the second mark M1."},{"type":"text","order":3,"title":"Simplify the expression","content":"Finally, we group and combine the like terms to simplify the expression completely:\n\n* **Cubic term ($y^3$):** $y^3$\n* **Quadratic terms ($y^2$):** $-y^2 + 2y^2 = y^2$\n* **Linear terms ($y$):** $-6y - 2y = -8y$\n* **Constant term:** $-12$\n\nPutting it all together, we get:\n\n$$y^3 + y^2 - 8y - 12$$\n\nThis final simplified expression earns the accuracy mark A1."}]}],"generatedAt":"2026-06-12T21:21:29.354Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"9bb1a3f1-e05d-47dd-bf6c-e9be2697680c","revisionId":"f2fc67e2-8e5f-4e49-a862-be64ef11d1ef","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"61d1ab25-d412-44af-9ec5-3728d33f2af6","label":"a","content":"Show that $(3x - 2)(x + 4)(2x - 1) = 6x^3 + 17x^2 - 26x + 8$ for all values of $x$.","markscheme":"- Expands two brackets to get $3x^2 + 10x - 8$ or $2x^2 + 7x - 4$ or $6x^2 - 7x + 2$ M1\n- Multiplies the resulting quadratic by the third linear bracket M1\n- Fully correct simplification to reach $6x^3 + 17x^2 - 26x + 8$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Expands any two brackets correctly to a quadratic expression.","code":"M"},{"id":"p2","marks":1,"description":"Multiplies their quadratic expansion by the third linear bracket correctly.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies fully to reach the target polynomial result.","code":"A"}],"finalAnswers":["6x^3 + 17x^2 - 26x + 8"],"generatedAt":"2026-05-06T14:01:08.562Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"expand_two_then_third","label":"Expand two brackets, then the third","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$(3x-2)(x+4)(2x-1)$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=expanded}{(3x^2+12x-2x-8)}}(2x-1)$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=quadratic}{(3x^2+10x-8)}}(2x-1)$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"expanded"},{"color":"orange","style":"text-color","anchor":"quadratic"}]},"order":0,"title":"Expand two brackets","description":"Multiply the first two brackets, then collect like terms. Correctly obtaining the quadratic earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anchor=quadratic}{(3x^2+10x-8)}}(2x-1)$"},{"latex":"$$\\color{@amber}{\\htmlData{anchor=split}{2x(3x^2+10x-8)-(3x^2+10x-8)}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=products}{(6x^3+20x^2-16x)-(3x^2+10x-8)}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"quadratic"},{"color":"amber","style":"text-color","anchor":"split"},{"color":"green","style":"text-color","anchor":"products"}]},"order":2,"title":"Multiply by each term","description":"Multiply the whole quadratic by 2x, then by -1. Writing this multiplication earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\htmlData{anch

Expanding Triple Brackets

Question 23

Expanding Triple Brackets Questions

Expanding Triple Brackets

Question 23

Expanding Triple Brackets Questions