AMC 10 2022b Problem 7

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots? $\textbf{(A)}\ 6 \textbf{(B)}\ 8 \textbf{(C)}\ 9 \textbf{(D)}\ 14 \textbf{(E)}\ 16$