Problem

Source: VIII International Festival of Young Mathematicians Sozopol 2017, Theme for 10-12 grade

Tags: algebra, functional equation



$f: \mathbb{R} \rightarrow \mathbb{R}$ is a function such that for $\forall x,y\in \mathbb{R}$ the equation $f(xy+x+y)=f(xy)+f(x)+f(y)$ is true. Prove that $f(x+y)=f(x)+f(y)$ for $\forall$ $x,y\in \mathbb{R}$.