Find all pairs of functions $f, g: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, we have $g(f(x+y))=f(x)+2(x+y) g(y)$
By taking $y=-x$, we have $g(f(0))=f(x)$ for all $x$. Consequently, $f$ is constant of the form $f(x)=c$ for all $x$. The condition then becomes $g(c)=c+2(x+y) g(y)$, so $g(c)-c=2(x+y) g(y)$. The left term is constant, so the right term must also be constant, which is only possible if $g$ is a constant equal to 0. The equation then becomes $0=f(x)$, so the only solution is $f(x)=g(x)=0$.