Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$.
I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with functional equations, finding the set the solutions is easy; it is proving that a certain set of solutions represents ALL solutions is what is difficult.
It is trivial that if $f$ is bounded $f\equiv0$.