Middle European Mathematical Olympiad 2010 Problem I-1

Find all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that for all $x,y\in \mathbb{R}$, we have $$f(x + y) + f(x)f(y) = f(xy) + (y + 1)f(x) + (x + 1)f(y).$$