Find the $f(x)$ if $f(f(x)+xf(y))=f(x)+xy$

Question

Find all $f:\Bbb R\to \Bbb R$,and such for any real number $x,y$ have $$f(f(x)+xf(y))=f(x)+xy$$

Let $x=y=0$ we have $$f(f(0))=f(0)$$ and I found a similar problem If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Answer 1

From $x=1$ we have $f(f(1)+f(y))=f(1)+y$, and so $f$ is injective. From $x=y=0$ we get $f(f(0))=f(0)$, and by injectivity $f(0)=0$. Next, from $y=0$ (keeping $x$ variable) we get $f(x)=f(f(x)+xf(0))=f(f(x))$, and again by injectivity $f(x)=x$. We can verify this is a solution, and thus the only one.