MEMO 2011 T1

Find all functions $f : \mathbb R \to \mathbb R$ such that the equality $$y^2f(x) + x^2f(y) + xy = xyf(x + y) + x^2 + y^2$$ holds for all $x, y \in \Bbb R$ , where $\Bbb R$ is the set of real numbers.