AMC 12 2022b Problem 4

For how many values of the constant $k$ will the polynomial $x^{2}+kx+36$ have two distinct integer roots?

$\textbf{(A) }6 \textbf{(B) }8 \textbf{(C) }9 \textbf{(D) }14 \textbf{(E) }16$