Solving a functional equation over the real number line

Question

Find all functions $f : R \to R$ which satisfy $$f(x)f(y) = f(xy) + xy$$

Answer 1

Let $y=1$, we have $$f(x)f(1)=f(x)+x$$ $$(f(1)-1)f(x)=x$$ $$f(x)=\frac{x}{f(1)-1}$$

So let $$f(x)=Ax$$ we have $$AxAy=Axy+xy$$ $$A^2-A-1=0$$ $$A=\frac{1\pm\sqrt{5}}{2}$$ $$f(x)=\frac{1\pm\sqrt{5}}{2}x$$