Middle European Mathematical Olympiad 2016 Problem T-2

Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f\colon \mathbb{R}\to \mathbb{R}$ such that $$f(x)f(y) = x f(f(y - x)) + x f(2x) + f(x^2)$$ holds for all real numbers $x$ and $y$.