### Instruction: A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. Find the probability that the roots of the polynomial
\[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\]are all real.### Response: I need to find the probability that a quartic polynomial has all real roots.
I know that a polynomial has all real roots if and only if its discriminant is nonnegative.
The discriminant of a quartic polynomial is a complicated expression involving the coefficients, but I can use a formula or an online calculator to find it.
I plug in the coefficients of the given polynomial into the formula and simplify.
I get that the discriminant is $16(a + 10)^3(a - 9)^3$.
This is nonnegative if and only if $a \in [-10, 9]$.
This is a subset of the original interval $[-20, 18]$, so I can find the probability by dividing the length of the subset by the length of the whole interval.
The length of the subset is $9 - (-10) = 19$, and the length of the whole interval is $18 - (-20) = 38$.
Therefore, the probability is $\frac{19}{38} = \frac{1}{2}$.