Question 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 $g(f(2)) = g(f(4)) = 0.$