Factorise: $2a^4 + a^2b^2 + ab^3 + b^4$

Question

Factorize : $$2a^4 + a^2b^2 + ab^3 + b^4$$ Here is what I did:

$$a^4+b^4+2a^2b^2+a^4-a^2b^2+ab^3+b^4$$ $$(a^2+b^2)^2+a^2(a^2-b^2)+b^3(a+b)$$ $$(a^2+b^2)^2+a^2(a+b)(a-b)+b^3(a+b)$$ $$(a^2+b^2)^2+(a+b)((a^2(a-b)) +b^3)$$ $$(a^2+b^2)^2+(a+b)(a^3-a^2b+b^3)$$

At this point I don't know what to do and am feeling that my direction is wrong. Please help me.
( Wolfram alpha says that the answer is $(a^2-a b+b^2) (2 a^2+2 a b+b^2)$ but how? )

Answer 1

Different Hint

Since your polynomial is homogeneous, this is equivalent to factoring the degree 4 univariate polynomial $2x^4+x^2+x+1.$

Answer 2

Hint

We look for a factorization on the form

$$2a^4 + a^2b^2 + ab^3 + b^4=(2a^2+\alpha ab+b^2)(a^2+\beta ab+b^2)$$

We find $\alpha=2,\beta=-1$.

Answer 3

After minar:

$2x^4+x^2+x+1= 2x^4+2x+x^2-x+1= 2x(x+1)(x^2-x+1)+x^2-x+1=(2x^2+2x+1)(x^2-x+1) $

Answer 4

Just Simple Factorisation: $2a^4+a^2b^2+ab^3+b^4$ $=2a^4+2ab^3+a^2b^2−ab^3+b^4$ $=2a(a^3+b^3)+a^2b^2−ab^3+b^4$ $=2a(a+b)(a^2-ab+b^2)+b^2(a^2-ab+b^2)$ $=(a^2-ab+b^2)(2a^2+2ab+b^2)$