Advanced Factoring Question

Hi all! I am actually trying to evaluate an integral but first I need to factor this expression.

$\displaystyle 30x^3+23x^2-29x+6$

Now since I have a solutions manual, I already know that the factorization of this is:

$\displaystyle (2x+3)(3x-1)(5x-2)$

So the rational zeros are $\displaystyle \frac{-3~}{2},~\frac{1}{3}~,\frac{2}{5}$. correct?

However, my solution manual does not explain the process of factoring this. How exactly do you factor this? My guess is that you can do a rational root test and find the rational zeros by trial and error with synthetic division. But, first of all, is it even possible to perform synthetic with fractions? And if so, and say you find that one rational root of some polynomial is $\displaystyle \frac{a}{b}$ for some $\displaystyle a,~b \in \mathbb{Z}.$ Does that ALWAYS mean that $\displaystyle (bx -a)$ is a factor of the expression?

Thanks for any help!

James