# Why Isn't The Answer 10000?

Number Theory Level 5

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

Over all such polynomials, what is the maximum possible value of $$N_f \times M_f$$?

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