Question:
Factorize each of the following expression:
(ax + by)2 + (bx − ay)2
Solution:
$(a x+b y)^{2}+(b x-a y)^{2}=a^{2} x^{2}+2 a b x y+b^{2} y^{2}+b^{2} x^{2}-2 a b x y+a^{2} y^{2}$
$=a^{2} x^{2}+b^{2} y^{2}+b^{2} x^{2}+a^{2} y^{2}$
$=\left(a^{2} x^{2}+a^{2} y^{2}\right)+\left(b^{2} x^{2}+b^{2} y^{2}\right)$ [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 $]$