2023 AMC 12B Problem 14

For how many ordered pairs (a,b) of integers does the polynomial x^3+ax^2+bx+6 have 3 distinct integer roots?

\textbf{(A)} 5 \qquad \textbf{(B)} 6 \qquad \textbf{(C)} 8 \qquad \textbf{(D)} 7 \qquad \textbf{(E)} 4

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