IZhO 2015 Problem 3

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f\left(x^{3}+y^{3}+x y\right)=x^{2} f(x)+y^{2} f(y)+f(x y)$$

for all $x, y \in \mathbb{R}$.