Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ for which $f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$

Question

Problem

Find all $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy

$$f(x^3)+f(y^3)=(x+y)(f(x^2)+f(y^2)-f(xy))$$ for all $x,y\in\mathbb{R}$.

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This is a contest math problem, and I have very little experience with functional equations. Thus I sadly have no progress to show. Admittedly, I'm posting this just to get some input on good strategies and solutions.