Is there an easy way to factorize this polynomial?

Question

I was solving some math exercises and I got stumped on the following problem:

"Write the expression $x^6+x^4+x^2y^2+y^4-y^6$ as the product of three factors"

I don't know how to start this. Can anyone help me factorize it?

Answer 1

Denote: $x^2=a, y^2=b$. Then: $$a^3+a^2+ab+b^2-b^3=(a^3-b^3)+(a^2+ab+b^2)=\\ (a-b)(a^2+ab+b^2)+(a^2+ab+b^2)=\\ (a-b+1)(a^2+ab+b^2)=(a-b+1)((a+b)^2-ab).$$ Can you substitute $x,y$ back and finish?

Answer 2

Hint $1$: factorise $x^6-y^6$.

Hint $2$: \begin{eqnarray*} (x^2+\alpha xy +y^2)(x^2-\alpha xy +y^2)=\cdots \end{eqnarray*}