mes 3$ multiplication grid:\n\n$$\n\\begin{array}{c|c|c|c}\n\\times & x^2 & +2x & -8 \\\\\n\\hline\nx & & & \\\\\n\\hline\n-5 & & & \n\\end{array}\n$$"},{"type":"text","order":3,"title":"Fill in the grid","content":"Now, systematically multiply the row headers by the column headers to fill in each cell of the grid:\n\n- Row $x$ multiplication:\n - $x \\times x^2 = x^3$\n - $x \\times 2x = 2x^2$\n - $x \\times -8 = -8x$\n\n- Row $-5$ multiplication:\n - $-5 \\times x^2 = -5x^2$\n - $-5 \\times 2x = -10x$\n - $-5 \\times -8 = 40$\n\nHere is our completed grid:\n\n$$\n\\begin{array}{c|c|c|c}\n\\times & x^2 & +2x & -8 \\\\\n\\hline\nx & x^3 & 2x^2 & -8x \\\\\n\\hline\n-5 & -5x^2 & -10x & 40 \n\\end{array}\n$$\n\nWriting down this step or showing equivalent multiplication terms earns you the second method mark M1."},{"type":"text","order":4,"title":"Combine like terms","content":"Collect all the terms from the grid and group the like terms together (the diagonals of the grid make this very easy!):\n\n$$\\begin{aligned} \\text{Expression} &= x^3 + (-5x^2 + 2x^2) + (-10x - 8x) + 40 \\\\ &= x^3 - 3x^2 - 18x + 40 \\end{aligned}$$\n\nThis fully simplified expression earns the final accuracy mark A1:\n\n$$x^3 - 3x^2 - 18x + 40$$"}]}],"generatedAt":"2026-06-13T00:00:03.655Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null},{"id":"aaeca733-68a4-4e50-a76a-c9538cf48643","revisionId":"6d2f5efd-13da-420d-9916-4315878c6533","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","parts":[{"id":"d17f63bc-b1fb-41ec-94b1-0875a30955d2","label":"a","content":"Show that $(4x + 1)(x - 5)(3x + 2) = 12x^3 - 49x^2 - 53x - 10$ for all values of $x$.","markscheme":"- Expands two brackets to get $4x^2 - 19x - 5$ or $12x^2 + 11x + 2$ or $3x^2 - 13x - 10$ M1\n- Multiplies the resulting quadratic by the third linear bracket M1\n- Fully correct simplification to reach $12x^3 - 49x^2 - 53x - 10$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Expands two brackets to get $4x^2 - 19x - 5$ or $12x^2 + 11x + 2$ or $3x^2 - 13x - 10$","code":"M"},{"id":"p2","marks":1,"description":"Multiplies the resulting quadratic by the third linear bracket","code":"M"},{"id":"p3","marks":1,"description":"Fully correct simplification to reach $12x^3 - 49x^2 - 53x - 10$","code":"A"}],"finalAnswers":["12x^3 - 49x^2 - 53x - 10"],"generatedAt":"2026-05-06T14:01:01.744Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"expand_in_stages","label":"Expand in stages","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=first_pair}{(4x+1)(x-5)}$"},{"latex":"$$\\htmlData{anchor=expanded_pair}{4x^2-20x+x-5}$"},{"latex":"$$\\htmlData{anchor=quadratic}{4x^2-19x-5}$"}],"connections":[{"to":"expanded_pair","from":"first_pair","color":"sky","style":"solid"},{"to":"quadratic","from":"expanded_pair","color":"orange","style":"solid"}]},"order":1,"title":"Expand two brackets","description":"Start with the first two brackets. Multiply every term in the first bracket by every term in the second, then collect the linear terms. Reaching the quadratic earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=product}{(4x^2-19x-5)(3x+2)}$"},{"latex":"$$\\htmlData{anchor=distributed}{3x(4x^2-19x-5)+2(4x^2-19x-5)}$"},{"latex":"$$\\htmlData{anchor=terms}{12x^3-57x^2-15x+8x^2-38x-10}$"}],"connections":[{"to":"distributed","from":"product","color":"sky","style":"solid"},{"to":"terms","from":"distributed","color":"green","style":"solid"}]},"order":2,"title":"Multiply third bracket","description":"Now multiply the whole quadratic by the third bracket. Showing this multiplication earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$12x^3+\\color{@purple}{(-57x^2+8x^2)}+\\color{@orange}{(-15x-38x)}-10$"},{"latex":"$$\

Expanding Triple Brackets

Question 37

Expanding Triple Brackets Questions

Expanding Triple Brackets

Question 37

Expanding Triple Brackets Questions