$$f(x) f(y) = f(x + y) + xy$$
I found by substitution that $f(0)=1$, and that either $f(1)$ or $f(-1)$ or both are equal to zero. I tried to do casework on the $f(1)$ and $f(-1)$, but don't know how to find every solution for $f$. Hints? Any way to prove that a solution is the only one?