How do you factor $\displaystyle 24x^3-96x^2+102x-30=6(2x-5)(x-1)(2x-1)$ without a computer? (Obviously, I used a computer).
Are you aware of the Rational Root Theorem?
For convenience I'm going to divide the polynomial by 6:
$\displaystyle 4x^3 - 16x^2 + 17x - 5 = 0$
Possible rational roots to this polynomial equation are:
$\displaystyle \pm \left ( 5,~\frac{5}{2},~\frac{5}{4},~1,~\frac{1}{2},~\frac {1}{4} \right )$
That gives you 12 possible rational roots that you can check.
If you don't know that one, then you're probably stuck with Cardano's method. (Cardano's method is about 1/2 way down the page.)
-Dan
If we just start with $\displaystyle 24x^3-96x^2+102x-30$, look for nice roots.
$\displaystyle x=1$ sort of pops out. So $\displaystyle (x-1)$ must be a factor.
Once we have one factor then by division we can reduce that to a quadratic which can factored.
I understand that it is not always possible to find a root that easily.