Suppose that $f: \mathbb R \rightarrow \mathbb R$ such that


$$f(x^3+y^3)=f(x+y)((f(x-y))^2+f(xy)),$$ for all $x,y$ real numbers. Is it true that the only solutions are $f(x)=0$ and $f(x)=x$? I did not come to any good result, but I think the solution should be difficult.



Is it true that the only solutions are $f(x)=0$ and $f(x)=x$?
No, the constant functions $$f=\frac{-1\pm\sqrt 5}{2}$$ are both solutions.


