Question:
Factorize:
$(a x+b y)^{2}+(b x-a y)^{2}$
Solution:
We have:
$(a x+b y)^{2}+(b x-a y)^{2}=\left[(a x)^{2}+2 \times a x \times b y+(b y)^{2}\right]+\left[(b x)^{2}-2 \times b x \times a y+(a y)^{2}\right]$
$=a^{2} x^{2}+2 a b x y+b^{2} y^{2}+b^{2} x^{2}-2 a b x y+a^{2} y^{2}$
$=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}$
$=\left(a^{2} x^{2}+b^{2} x^{2}\right)+\left(a^{2} y^{2}+b^{2} y^{2}\right)$
$=x^{2}\left(a^{2}+b^{2}\right)+y^{2}\left(a^{2}+b^{2}\right)$
$=\left(a^{2}+b^{2}\right)\left(x^{2}+y^{2}\right)$