Question
Factorise the following:$\ (y^2 - 3y)(y^2 - 3y + 7) + 10$
✓
Answer
$(y^2 - 3y)(y^2 - 3y + 7) + 10$
$= a(a + 7) + 10[$taking $(y^2 - 3y) = a]$
$= a^2 + 7a + 10$
$= a^2+ 5a + 2a + 10$
$= a(a + 5) + 2(a + 5)$
$= (a + 5)(a + 2)$
$= (y^2 - 3y + 5)(y^2 - 3y + 2)$
$= (y^2 - 3y + 5)(y^2 - 2y - y + 2)$
$= (y^2 - 3y + 5)[y(y - 2) - 1(y - 2)]$
$= (y^2 - 3y + 5)[(y - 2)(y - 1)]$
$= (y - 1)(y - 2)(y^2 - 3y + 5).$
$= a(a + 7) + 10[$taking $(y^2 - 3y) = a]$
$= a^2 + 7a + 10$
$= a^2+ 5a + 2a + 10$
$= a(a + 5) + 2(a + 5)$
$= (a + 5)(a + 2)$
$= (y^2 - 3y + 5)(y^2 - 3y + 2)$
$= (y^2 - 3y + 5)(y^2 - 2y - y + 2)$
$= (y^2 - 3y + 5)[y(y - 2) - 1(y - 2)]$
$= (y^2 - 3y + 5)[(y - 2)(y - 1)]$
$= (y - 1)(y - 2)(y^2 - 3y + 5).$
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