jmc

Let $r_1$ and $r_2$ be the roots of this polynomial. Therefore, $r_1+r_2=13$ and $r_1{r}_2=4$. Notice that $r_1^2+2r_1{r}_2+r_2^2=169$. This means that the sum of the squares of the roots can be obtained by subtracting the term containing the product of $r_1$ and $r_2$, so $r_1^2+r_2^2=169-2(4)=\boxed{161}$.