A real number ‘$a$’ is chosen randomly and uniformly from the interval $\left[-20,18\right]$ the probability that the roots of the polynomial $x^4+2a{x}^3+(2a-2)x^2+(-4a+3)x-2$ are all real is

By rational root theorem turn the given polynomial into a degree = 2 polynomial with
rational root
x = 1, -1, 2, -2 are all possible root
So, x -2, x=1. Polynomial is equal to 0

So, polynomial becomes $(x-1)(x+2)\left(x^2+(2a-1)x+1\right)$

So, we only need to focus on the quadratic
$(2a-1{)}^2-4\ge 0\Rightarrow n(S)=38;n(E)=36;\therefore \mathrm{Pr}\text{ obability }=\frac{36}{38}=\frac{18}{19}$