Find all function $f:\mathbb{R}\to\mathbb{R}$ such that $$f(f(x)+y)=f(x^2-y)+4f(x)y, \forall x, y\in\mathbb{R}$$
By substitution $y\to\frac{x^2-f(x)}{2}$, $$f(x)(x^2-f(x))=0$$. Now I want to prove that if there os some $x_0\neq0$ such that $f(x_0)=0$, then $f(x)=0,\forall x\in\mathbb{R}$
Substituting $x\to x_0$, $$f(x_0^2-y)=f(y)\implies f(x_0^2)=f(0)=0$$. Then by induction $f(x_0^{2^n})=0$. But I can't finish that. Can anyone help me?