Problem

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2016

Tags: function, functional equation, algebra



Find all functions $f : \mathbb{Q} \rightarrow \mathbb{R}$ such that: $a)$ $f(1)+2>0$ $b)$ $f(x+y)-xf(y)-yf(x)=f(x)f(y)+f(x)+f(y)+xy$, $\forall x,y \in \mathbb{Q}$ $c)$ $f(x)=3f(x+1)+2x+5$, $\forall x \in \mathbb{Q}$