The roots of the equation $2x^2 - 5x - 4 = 0$ can be written in the form $x = \frac{m \pm \sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers with a greatest common divisor of 1. What is the value of $n$?
This problem is an application of the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Using the quadratic formula, we find that $x = \frac{5 \pm \sqrt{25 +32}}{4} = \frac{5 \pm \sqrt{57}}{4}$. Since $4$ and $57$ are relatively prime, $n = \boxed{57}$.