the second bracket $(y^2 + 4y + 4)$:\n\n$$y(y^2 + 4y + 4) - 3(y^2 + 4y + 4)$$\n\n*This multiplication setup earns you the second method mark (\\text{M1}).*"},{"type":"text","order":3,"title":"Expand and simplify","content":"Now we expand the distributed terms:\n\n$$\\begin{aligned} y(y^2 + 4y + 4) &= y^3 + 4y^2 + 4y \\\\ -3(y^2 + 4y + 4) &= -3y^2 - 12y - 12 \\end{aligned}$$\n\nCombining these results gives:\n\n$$y^3 + 4y^2 + 4y - 3y^2 - 12y - 12$$\n\nNow, collect the like terms:\n- Cubic term: $y^3$\n- Quadratic terms: $4y^2 - 3y^2 = y^2$\n- Linear terms: $4y - 12y = -8y$\n- Constant term: $-12$\n\nCombining these, we get our final simplified answer:\n\n$$y^3 + y^2 - 8y - 12$$\n\n*Correctly obtaining this simplified expression earns you the accuracy mark (\\text{A1}).*"}]},{"id":"ff8ffedd-80d2-4347-b6e8-db339ae85fde:expand_linear_pair_first","label":"Expand Linear Pair First","steps":[{"type":"text","order":0,"title":"Rewrite the expression","content":"First, let's write out the squared term to see all three linear factors clearly:\n\n$$(y - 3)(y + 2)^2 = (y - 3)(y + 2)(y + 2)$$\n\nTo use the **Expand Linear Pair First** approach, we will begin by multiplying the first two brackets, $(y - 3)$ and $(y + 2)$, together."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=pair}{(y - 3)(y + 2)}$"},{"latex":"$$= y(y + 2) - 3(y + 2)$"},{"latex":"$$= \to \begin{aligned} y^2 &+ 2y \\ &- 3y - 6 \n\\end{aligned}$"},{"latex":"$$= \framebox{\\htmlData{anchor=quadratic}{y^2 - y - 6}}$"}],"highlights":[{"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."

Expanding Triple Brackets Questions

Expanding Triple Brackets Questions

Expanding Triple Brackets Questions

Expanding Triple Brackets Questions