2023 AMC 12B Problem 14

By Vieta, the three distinct integer roots multiply to $-6.$ The sets of three distinct integers with product $-6$ are $\{1,2,-3\},$ $\{1,-2,3\},$ $\{-1,2,3\},$ $\{-1,-2,-3\},$ and $\{1,-1,6\}.$ Each set determines $a=-(p+q+r)$ and $b=pq+qr+rp,$ and all five give different pairs, so there are $5$ ordered pairs $(a,b).$ Thus, the correct answer is A.