jmc

To find the coefficient of $x^2{y}^2$, we can find the coefficient in each of the 4th powers. These are the only things that can contribute $x^2{y}^2$ terms to the final expansion.

The expansion of $(x+y{)}^4$ will have a coefficient of $\left(\genfrac{}{}{0ex}{}{4}{2}\right)$ for $x^2{y}^2$, by the Binomial Theorem.

The expansion of $(x+2y{)}^4$ will have a coefficient of $2^2\left(\genfrac{}{}{0ex}{}{4}{2}\right)$ for $x^2{y}^2$.

Thus the coefficient of $x^2{y}^2$ in the expansion of $(x+y{)}^4+(x+2y{)}^4$ will be $\left(\genfrac{}{}{0ex}{}{4}{2}\right)+4\left(\genfrac{}{}{0ex}{}{4}{2}\right)=\boxed{30}$.