2018 AIME II Problem 6

A real number a is chosen randomly and uniformly from the interval [-20, 18]. The probability that the roots of the polynomial

x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2

are all real can be written in the form \dfrac{m}{n}, where m and n are relatively prime positive integers. Find m + n.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.