Factorise:

We have:

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

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

$=a x(b x-a y)+b y(a y-b x)$

$=a x(b x-a y)-b y(b x-a y)$

$=(b x-a y)(a x-b y)$

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