AMC 12 2008a Problem 19

In the expansion of $$\left(1 + x + x^2 + \cdots + x^{27}\right)\left(1 + x + x^2 + \cdots + x^{14}\right)^2,$$ what is the coefficient of $x^{28}$?

$\mathrm{(A)}\ 195 \mathrm{(B)}\ 196 \mathrm{(C)}\ 224 \mathrm{(D)}\ 378 \mathrm{(E)}\ 405$