I tried using the Rational Root Theorem on $x^3-6x^2+4x-5=0$. However, I could not find a rational root of the cubic. When I plugged the cubic into Wolfram alpha, it yielded a very messy real solution:
How do you tell whether the RRT doesn't work on a particular cubic of the form $ax^3+bx^2+cx+d=0$? How do you tell whether $ax^3+bx^2+cx+d=0$ doesn't have any rational roots?