$f:\Bbb{R} \to \Bbb{R}, f(xf(y)+y)=yf(x)+f(y)$
\begin{align} P(x, 0): \; & f(xf(0))=f(0). \\ & \text{if } f(0) \neq 0: \\ P\left(\frac{x}{f(0)}, 0\right): \; & f(x)=f(0), f \equiv c. \\ \Rightarrow \; & c=(y+1)c. (\otimes) \\ \therefore \; & f(0)=0. \\ \ \\ & \text{if } \exists t \text{ s.t. } f(t)=0, t\neq0: \\ P(x, t): \; & 0=tf(x), f \not\equiv c \Rightarrow t=0. (\otimes) \\ \therefore \; & f(t)=0 \Leftrightarrow t=0. \\ \ \\ P(x, x): \; & f(x(f(x)+1))=f(x)(x+1). \end{align}
Then I got stuck... Can someone help me?
(This problem is from some Olympiad problem book.)