MathNet 2013 Question 93

Find all pairs $(f, h)$ of functions $f$ and $h$, $f : \mathbb{R} \to \mathbb{R}$, $h : \mathbb{R} \to \mathbb{R}$, such that the equality $f(x^2 + y h(x)) = x h(x) + f(xy)$ holds for all real $x$ and $y$.