IR-TST 2012 Problem 8

Let $g(x)$ be a polynomial of degree at least $2$ with all of its coefficients positive. Find all functions $f:\mathbb R^+ \longrightarrow \mathbb R^+$ such that $$f(f(x)+g(x)+2y)=f(x)+g(x)+2f(y) \forall x,y\in \mathbb R^+.$$

Proposed by Mohammad Jafari