We know that for two real numbers $a,b$ and two random variables $X,Y$ we have that $E(a X + b Y ) = a E(X) + b E(Y)$. Under what conditions is it also true that for any three random variables $X,Y,Z$, we have that $E\bigl(X (Y + Z)\bigr) = E(X Y + X Z) = E(X Y) + E(X Z)$?
In this equation you are allowed to assume that there exist joint distributions for $X,Y$ and $X,Z$ but not necessarily for $X,Y,Z$. So, as per Michael's answer below, the question would seem to reduce to when a joint distribution exists given distributions for $X,Y$ and $X,Z$.