Answer to Question #43622 in Algebra for ruchi

Question

factorize
ab(x^2+y^2) - xy(a^2+b^2)

Answer 1

Distribute ab over x²+y²: ab(x^2+y^2)=abx^2+aby^2: abx^2+aby^2-xy(a^2+b^2).
Distribute -xy over a²+b²: -xy(a^2+b^2)=-(a^2xy)-b^2xy: abx^2+aby^2+[-(a^2xy)-b^2xy].
Factor pairs of terms in
abx^2+aby^2-a^2xy-b^2xy by grouping:

Factor terms by grouping. abx^2+aby^2-a^2xy-b^2xy=(abx^2-a^2xy)+(aby^2-b^2xy)=by(ay-bx)-ax(ay-bx):
by(ay-bx)-ax(ay-bx).

Factor common terms from by(ay-bx)-ax(ay-bx).
Factor ay-bx from by(ay-bx)-ax(ay-bx): (ay-bx)(by-ax).

Answer. (ay-bx)(by-ax).