AMC 12 2023b 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 \textbf{(B)}\ 6 \textbf{(C)}\ 8 \textbf{(D)}\ 7 \textbf{(E)}\ 4