smc

Let $A=\left(1+x+x^2+\cdots +x^{14}\right)$ and $B=\left(1+x+x^2+\cdots +x^{27}\right)$. We are expanding $A\cdot A\cdot B$. Since there are $15$ terms in $A$, there are $15^2=225$ ways to choose one term from each $A$. The product of the selected terms is $x^n$ for some integer $n$ between $0$ and $28$ inclusive. For each $n\ne 0$, there is one and only one $x^{28-n}$ in $B$. For example, if I choose $x^2$ from $A$ , then there is exactly one power of $x$ in $B$ that I can choose; in this case, it would be $x^{24}$. Since there is only one way to choose one term from each $A$ to get a product of $x^0$, there are $225-1=224$ ways to choose one term from each $A$ and one term from $B$ to get a product of $x^{28}$. Thus the coefficient of the $x^{28}$ term is $224\Rightarrow \boxed{224}$.