2014 AMC 12A 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\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.