$\displaystyle 20x^3-68x^2+69x-18=0$ Other than trial and error method... is there any way to factorize this faster?
The fastest method is to use a Computer Algebra System...
The one I used got
$\displaystyle 20x^3 - 68x^2 + 69x - 18 =\pm(2x - 3)^2(5x - 2)$.
If you were going to use the Factor and Remainder Theorems, you would need to substitute $\displaystyle x = \frac{2}{5}$ or $\displaystyle x = \frac{3}{2}$ and find that the polynomial equals 0.