Question 18
Let $g$ be a polynomial of degree at least $2$ with nonnegative coefficients. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that for every $x, y \in \mathbb{R}^{+}$
$$
f(f(x) + g(x) + 2y) = f(x) + g(x) + 2f(y).
$$
$$
f(f(x) + g(x) + 2y) = f(x) + g(x) + 2f(y).
$$