How can $f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$ be factorized into a product of two polynomials? Let $x,y$ be 2 coprime integers. I assume the following polynomial:$$f(x,y)= x^4+x^3y+x^2y^2+xy^3+y^4$$ is not irreducible. So there must be at least 2 other polynomials of degree $\leq 4$ such that: $$f(x,y)=g(x,y)h(x,y)$$
How can one find $g(x,y)$ and $h(x,y)$?

According to marty cohen, we have
\begin{align*}
r^4+r^3+r^2+r+1&=\left(r^2+\frac{1}{2}r+1\right)^2-\frac{5}{4}r^2\\
&=\left(r^2+\frac{1+\sqrt{5}}{2}r+1\right)\left(r^2+\frac{1-\sqrt{5}}{2}r+1\right)\\
\end{align*}
Then
$$x^4+x^3y+x^2y^2+xy^3+y^4=\left(x^2+\frac{1+\sqrt{5}}{2}xy+y^2\right)\left(x^2+\frac{1-\sqrt{5}}{2}xy+y^2\right)$$
