the problem is:
It says something about guessing and checking, but how do I know if the number I pick works?
rational roots theorem to check for rational zeros. the roots here might not be rational since they ask for bounds rather than the actual zeros, but who knows. start by plugging in the factors (+/-) of 60 to see if you get zero when you evaluate it. if you do, great, if not, try picking numbers close to that that get you as close to zero as possible from both sides, that will get you the bounds. if you actually find a rational zero, call it a, then you can divide the polynomial by (x - a) and it will reduce it to a quadratic where you can find the other zeros.
The factors of 24 are so you try in turn untill you find one that works.
Once you have found a root r, you know that is a factor, and so you can divide the cubic through by the factor to find quadratic such that:
and carry on from there