Question

Factorise:

$(a^2+b^2-4c^2{)}^2-4a^2{b}^2$

Answer 1

$(a^2+b^2-4c^2{)}^2-4a^2{b}^2=(a^2+b^2-(2c{)}^2{)}^2-(2ab{)}^2,\text{using identity}(a^2-b^2)=(a+b)(a-b)=(a^2+b^2-(2c{)}^2-2ab\left)\right(a^2+b^2-(2c{)}^2+2ab)=(a^2+b^2-2ab-(2c{)}^2\left)\right(a^2+b^2+2ab-\left(2c{)}^2\right)=\left(\right(a-b{)}^2-\left(2c{)}^2\right)\left(\right(a+b{)}^2-\left(2c{)}^2\right)=\left[\right(a-b-2c\left)\right(a-b+2c\left)\right]\left[\right(a+b-2c\left)\right(a+b+2c\left)\right]=(a-b-2c)(a-b+2c)(a+b-2c)(a+b+2c)$