Question 4
Let $c\gt 0$ be a given positive real and $\mathbb R_{\gt 0}$ the set of all positive reals. Find all functions $f:\mathbb R_{\gt 0}\to\mathbb R_{\gt 0}$ such that
$$f((c+1)x+f(y))=f(x+2y)+2cx$$
for all $x,y\in\mathbb R_{\gt 0}$.
$$f((c+1)x+f(y))=f(x+2y)+2cx$$
for all $x,y\in\mathbb R_{\gt 0}$.