What is the right approach to use for factoring a rational inequality Here is an example:
$\frac{x^2 - 3x + 2}{x + 1} -5 > 0$
My approach would be to factor, find the undefined areas and the zeros, and then pick some points in the intervals left to see what I find. I'm not really sure if that's the right way to go, and to find the zeros I would just throw an equals sign in instead of the inequality and find the roots. Is this the right approach, and are there any techniques to improve it?

This is not a general approach, but in this example $x=-1$ is 'special', so split the consideration into $I_- = (-\infty,-1)$ and $I_+ = (-1, +\infty)$.
On $I_+$, we have $x^2 - 3x + 2 - 5 (x + 1) = x^2-8x-3 > 0$. The factors are $4 \pm \sqrt{19}$, hence if $x \in I_+$, then $\frac{x^2 - 3x + 2}{x + 1} -5 > 0$ iff $x > 4 + \sqrt{19}$ or $x < 4 - \sqrt{19}$.
On $I_-$, we have $x^2 - 3x + 2 - 5 (x + 1) =x^2-8x-3 < 0$. From the previous reasoning, we have that if $x \in I_-$, then $\frac{x^2 - 3x + 2}{x + 1} -5 \leq 0$.
Hence $\frac{x^2 - 3x + 2}{x + 1} -5 > 0$ iff $ x \in (-1,4 - \sqrt{19}) \cup (4 + \sqrt{19}, +\infty)$.
