Consider the quadratic expression $13x^2 + nx - 17.$ For certain values of $n,$ it may be factored into a product of two linear polynomials,

The factored polynomial is of the form $(ax + b)(cx + d)$. Since 13 and 17 are primes, either a or c is 13, and either b or d is 17 or -17.

Listing all possibilities:

$\begin{array}{rcl} (13x - 1)(x + 17) &=& 13x^2 + 220x - 17\\ (13x + 1)(x - 17) &=& 13x^2 -220x - 17\\ (13x + 17)(x - 1) &=& 13x^2 + 4x - 17\\ (13x - 17)(x + 1) &=& 13x^2 - 4x - 17 \end{array}$

The possible values of n are -220, -4, 4, 220.