  The purpose of the present paper is to solve (under some assumption on the
domain) the equation $$ g(x+y)-g(x)-g(y)=xf(y)+yf(x). $$ After determining the
general solutions, we will investigate the so--called alien solutions. %More
precisely, we will examine the cases when the above equation implies that
%$g(x+y)=g(x)+g(y)$ and $xf(y)+yf(x)=0$. %Concerning this, necessary and
sufficient conditions will be provided. Finally, we will discuss the real
solutions of the following related functional inequality: $$
g(x+y)-g(x)-g(y)\geq xf(y)+yf(x). $$
