Question

Factorise $4x^4+9y^4+6x^2{y}^2$.

Answer 1

Taking $a=\sqrt{2}x,b=\sqrt{3}y$, we obtain
$a^4+a^2{b}^2+b^4=4x^4+6x^2{y}^2+9y^4$
We can use factorisation we have just proved. Thus
$4x^4+6x^2{y}^2+9y^4=a^4+a^2{b}^2+b^4=(a^2+ab+b^2)(a^2-ab+b^2)$
Putting back the value of $a$ and $b$, we obtain
$4x^4+6x^2{y}^2+9y^4=(2x^2-\sqrt{6}xy+3y^2)(2x^2+\sqrt{6}xy+3y^2)$.