Why Isn't The Answer 10000?

Consider all polynomials f(x)f(x) with integer coefficients and degree at most 100. There are NfN_f distinct integer values for which f(n)=2f(n) = 2, and MfM_f distinct integer values for which f(m)=2f(m)=-2.

Over all such polynomials, what is the maximum possible value of Nf×MfN_f \times M_f?

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