Question 6
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18].$ The probability that the roots of the polynomial $$
\begin{aligned} &x^4 + 2ax^3 + (2a - 2)x^2 \\ &\quad {}+ (-4a + 3)x - 2 \end{aligned}
$$ are all real can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
\begin{aligned} &x^4 + 2ax^3 + (2a - 2)x^2 \\ &\quad {}+ (-4a + 3)x - 2 \end{aligned}
$$ are all real can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$