Question
Factorise the following:$\ 12 - (y + y^2)(8 - y - y^2)$
✓
Answer
$12 - (y + y^2)(8 - y - y^2)$
$= 12 - a(8 - a)[$Taking $y + y^2 = a]$
$= 12 - 8a + a^2$
$= 12 - 6a - 2a + a^2$
$= 6(2 - a) - a(2 - a)$
$= (2 - a)(6 - a)$
$= [2 - (y + y^2)][6 - (y + y^2)]$
$= (2 - y - y^2)(6 - y - y^2)$
$= (2 - 2y + y - y^2)(6 - 3y + 2y - y^2)$
$= [2(1 - y) + y(1 - y)][3(2 - y) + y(2 - y)]$
$= [(1 - y)(2 + y)][(2 - y)(3 + y)]$
$= (1 - y)(2 + y)(2 - y)(3 + y)$
$= (y - 1)(y + 2)(y - 2)(y + 3).$
$= 12 - a(8 - a)[$Taking $y + y^2 = a]$
$= 12 - 8a + a^2$
$= 12 - 6a - 2a + a^2$
$= 6(2 - a) - a(2 - a)$
$= (2 - a)(6 - a)$
$= [2 - (y + y^2)][6 - (y + y^2)]$
$= (2 - y - y^2)(6 - y - y^2)$
$= (2 - 2y + y - y^2)(6 - 3y + 2y - y^2)$
$= [2(1 - y) + y(1 - y)][3(2 - y) + y(2 - y)]$
$= [(1 - y)(2 + y)][(2 - y)(3 + y)]$
$= (1 - y)(2 + y)(2 - y)(3 + y)$
$= (y - 1)(y + 2)(y - 2)(y + 3).$
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