2020 AIME I Problem 11

For integers a,b,c and d, let f(x)=x^2+ax+b and g(x)=x^2+cx+d. Find the number of ordered triples (a,b,c) of integers with absolute values not exceeding 10 for which there is an integer d such that g(f(2))=g(f(4))=0.

Leading zeroes must be inputted, so if your answer is 34, then input 034

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