Factorize:

We have:

$(a x+b y)^{2}+(b x-a y)^{2}=\left[(a x)^{2}+2 \times a x \times b y+(b y)^{2}\right]+\left[(b x)^{2}-2 \times b x \times a y+(a y)^{2}\right]$

$=a^{2} x^{2}+2 a b x y+b^{2} y^{2}+b^{2} x^{2}-2 a b x y+a^{2} y^{2}$

$=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}$

$=\left(a^{2} x^{2}+b^{2} x^{2}\right)+\left(a^{2} y^{2}+b^{2} y^{2}\right)$

$=x^{2}\left(a^{2}+b^{2}\right)+y^{2}\left(a^{2}+b^{2}\right)$

$=\left(a^{2}+b^{2}\right)\left(x^{2}+y^{2}\right)$