AMC 12 2014a Problem 19

There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $$5x^2+kx+12=0$$ has at least one integer solution for $x$. What is $N$?

$\textbf{(A) }6 \textbf{(B) }12 \textbf{(C) }24 \textbf{(D) }48 \textbf{(E) }78$