Reduce me non-constantly

What are the last three digits of the product of absolute values of all integers \(n,\) such that the polynomial \[f_n(x)=x^5+nx^3-3x^2-9\] can be expressed as a product of two non-constant polynomials with integer coefficients?

Details and assumptions

The last three digits of the number 1023 are 023. You can type in your answer as 023 or 23.

If you think that the integers \( 2, -2, -3, -4 \) satisfy the conditions, then the product of absolute values of all these integers is \( 2 \times 2 \times 3 \times 4 \).

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