How many polynomials?

For every positive integer n,n, consider all monic polynomials f(x)f(x) with integer coefficients, such that for some real number aa x[f(x+a)f(x)]=nf(x)x\left[ f(x+a)-f(x) \right]=nf(x) Find the largest possible number of such polynomials f(x)f(x) for a fixed n<1000.n<1000.

Details and assumptions

A polynomial is monic if its leading coefficient is 1. For example, the polynomial x3+3x5 x^3 + 3x - 5 is monic but the polynomial x4+2x36 -x^4 + 2x^3 - 6 is not.

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