Question
Factorise the following$: \ (a^2 - 2a)^2 - 23(a^2 - 2a) + 120$
✓
Answer
$(a^2 - 2a)^2 - 23(a^2 - 2a) + 120$
$= (a^2 - 2a)^2 - 15(a^2 - 2a) - 8(a^2 - 2a) + 120$
$= (a^2 - 2a)(a^2 - 2a - 15) - 8(a^2 - 2a - 15)$
$= (a^2 - 2a - 15)(a^2 - 2a - 8)$
$= (a^2 - 5a + 3a - 15)(a^2 - 4a + 2a - 8)$
$= [a(a - 5) + 3(a - 5)][a(a - 4) + 2(a - 4)]$
$= [(a - 5)(a + 3)][(a - 4)(a + 2)]$
$= (a - 5)(a + 3)(a - 4)(a + 2)$
$= (a + 2)(a + 3)(a - 4)(a - 5).$
$= (a^2 - 2a)^2 - 15(a^2 - 2a) - 8(a^2 - 2a) + 120$
$= (a^2 - 2a)(a^2 - 2a - 15) - 8(a^2 - 2a - 15)$
$= (a^2 - 2a - 15)(a^2 - 2a - 8)$
$= (a^2 - 5a + 3a - 15)(a^2 - 4a + 2a - 8)$
$= [a(a - 5) + 3(a - 5)][a(a - 4) + 2(a - 4)]$
$= [(a - 5)(a + 3)][(a - 4)(a + 2)]$
$= (a - 5)(a + 3)(a - 4)(a + 2)$
$= (a + 2)(a + 3)(a - 4)(a - 5).$
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