Factorize each of the following expression:

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

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

$=\left(a^{2} x^{2}+a^{2} y^{2}\right)+\left(b^{2} x^{2}+b^{2} y^{2}\right)$      [Regrouping the expressions]

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

$=\left(a^{2}+b^{2}\right)\left(x^{2}+y^{2}\right)$         [Taking $\left(x^{2}+y^{2}\right)$ as the common factor $]$