How to factor a quartic polynomial in two variables?

Question

How can I factor the following polynomial, $$ a^4+2a^3b+3ab^2+2ab^3+b^4$$

Answer 1

$$a^4+2a^3b+3a^2b^2+2ab^3+b^4=a^4+b^4+3a^2b^2+2ab(a^2+b^2)=$$ $$(a^2+b^2)^2+a^2b^2+2ab(a^2+b^2)=(a^2+ab+b^2)^2.$$

Answer 2

In the $(1,2,3,2,1)$ pattern you may recognize the convolution between $(1,1,1)$ and $(1,1,1)$. Since the given polynomial is homogeneous, it follows it is the square of $\color{red}{1}a^2+\color{red}{1}ab+\color{red}{1}b^2$.
In particular $$ a^4+2a^3 b+3a^2 b^2+2a b^3+b^4 = \left(\frac{a^3-b^3}{a-b}\right)^2 $$

and in a similar fashion $$\begin{eqnarray*} a^6 + 2 a^5 b + 3 a^4 b^2 + 4 a^3 b^3 + 3 a^2 b^4 + 2 a b^5 + b^6 &=& (a^3+a^2 b+a b^2+b^3)^2 \\ &=&\left(\frac{a^4-b^4}{a-b}\right)^2\\&=&(a+b)^2 (a^2+b^2)^2. \end{eqnarray*}$$