BALTIC-WAY 2010 Problem 5

Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

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

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