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If f(x) = x^4 - 3x^3 - 9x^2 + 4 ,
for how many real numbers does f(k) = 2?

The answer is 4, and I answered it correctly. However, I answered it mostly on intuition and would like to know a more sound method of eliminating imaginary zeros from the resulting equation x^4 - 3x^3 - 9x^2 + 2 = 0. Since the rational zeros (-2,2,-1,1) are not solutions, and since the amount of positive or negative real zeros is 2 OR 0, I would just like to know how to eliminate the options of 2 or 4 imaginary zeros. Thanks guys.

EDIT: Never mind I've got it, silly me. If x were imaginary, then the term 3x^3 would have an imaginary component which would make it impossible for the result to be correct.