The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $f(x) f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Find all possible functions $f$.

Question

The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$f(x) f(y) = f(x + y) + xy$$ for all real numbers $x$ and $y$. Find all possible functions $f$.

Answer 1

I cannot write the whole answer. Here are hints for you.

Compute $f(0)$ by taking $x=y=0$ If $f(0)=0$ then choose $y=0$... If $f(0)=1$ then find $a$ such that $f(a)=0$ ( $a \in \{1,-1\}$). Then choose $y=a$ deduces the results.

Thanks for the comments. There are 2 solutions.