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 $m + n$.
The polynomial we are given is rather complicated, so we could use [Rational Root Theorem](https://artofproblemsolving.com/wiki/index.php/Rational_Root_Theorem) to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, $x = 1, -1, 2, -2$ are all possible rational roots. Upon plugging these roots into the polynomial, $x = -2$ and $x = 1$ make the polynomial equal 0 and thus, they are roots that we can factor out.
The polynomial becomes:
$(x - 1)(x + 2)(x^2 + (2a - 1)x + 1)$
Since we know $1$ and $-2$ are real numbers, we only need to focus on the quadratic.
We should set the discriminant of the quadratic greater than or equal to 0.
$(2a - 1)^2 - 4 \geq 0$.
This simplifies to:
$a \geq \dfrac{3}{2}$
or
$a \leq -\dfrac{1}{2}$
This means that the interval $\left(-\dfrac{1}{2}, \dfrac{3}{2}\right)$ is the "bad" interval. The length of the interval where $a$ can be chosen from is 38 units long, while the bad interval is 2 units long. Therefore, the "good" interval is 36 units long.
$\dfrac{36}{38} = \dfrac{18}{19}$
$18 + 19 = \boxed{037}$
~First