Question 29
Find all even functions $g: \mathbb{R} \rightarrow \mathbb{R}$ for which there exists a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every $x, y \in \mathbb{R}$
$$
g(f(x)+y)=g(x)+g(y)+y f(x+f(x))
$$
$$
g(f(x)+y)=g(x)+g(y)+y f(x+f(x))
$$