Question
Factorise$ : (a^2+ b^2- 4c^2)^2- 4a^2b^2$
✓
Answer
$\left(a^2+b^2-4 c^2\right)^2-4 a^2 b^2$
$=\left(a^2+b^2-4 c^2\right)^2-(2 a b)^2$
$=\left(a^2+b^2-4 c^2-2 a b\right)\left(a^2+b^2-4 c^2+2 a b\right) $
$\left[\because a^2-b^2=(a\right.+b)(a-b)]$
$=\left(a^2+b^2-2 a b-4 c^2\right)\left(a^2+b^2+2 a b-4 c^2\right)$
$=\left[(a-b)^2-(2 c)^2\right]\left[(a+b)^2-(2 c)^2\right]$
$=(a-b+2 c)(a-b-2 c)(a+b+2 c)(a+b-2 c)$
$=\left(a^2+b^2-4 c^2\right)^2-(2 a b)^2$
$=\left(a^2+b^2-4 c^2-2 a b\right)\left(a^2+b^2-4 c^2+2 a b\right) $
$\left[\because a^2-b^2=(a\right.+b)(a-b)]$
$=\left(a^2+b^2-2 a b-4 c^2\right)\left(a^2+b^2+2 a b-4 c^2\right)$
$=\left[(a-b)^2-(2 c)^2\right]\left[(a+b)^2-(2 c)^2\right]$
$=(a-b+2 c)(a-b-2 c)(a+b+2 c)(a+b-2 c)$
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