How to solve the functional equation $ f(x^2+xf(y))= xf(y)$

Question

Hello please how to find all the functions $f:\mathbb{R}\to \mathbb{R}$ such that $$ f(x^2+xf(y))= xf(y)$$ I see that $f(0)=0$ but how to do after

Answer 1

$x=0$ gives use $f(0)=0$. Then with $x=-f(y)$, $$ -f(y)^2=f(x^2+xf(y))=f(0)=0.$$ We conclude $f\equiv 0$.