Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation $f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $
Let $$P(x,y): f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)$$ $$P(0,1): f(f(1))=1+f(0)$$ $$P(1,0): f(1+f(0))= f(0)+f(1)$$ Hence, from $P(0,1)$ and $P(1,0)$, we get $$f(1+f(0))=f(0)+f(1) \implies f(0)=f(f(0))=0$$ Now, $$P(0, b): f(f(b^2))=b^2 \quad \forall b \in \mathbb{R^+}$$ What do I do after.