Find the number of cubic polynomials p(x)=x^{3}+a x^{2}+b x+c, where a, b, and c are integers in \{-20,-19,-18, \ldots, 18,19,20\}, such that there is a unique integer m \neq 2 with p(m)=p(2).
Find the number of cubic polynomials p(x)=x^{3}+a x^{2}+b x+c, where a, b, and c are integers in \{-20,-19,-18, \ldots, 18,19,20\}, such that there is a unique integer m \neq 2 with p(m)=p(2).