TITLE: Different ways to factor
QUESTION [8 upvotes]: I'm interested to know some different methods to factor equations of the form $ax^2+bx+c$, where $a \ne 0$, other than pure guess and check.

REPLY [4 votes]: Another option:
$$a\,{{x}^{2}}+bx+c$$
$$\downarrow$$
$$n\,{{x}^{2}}-2knx+{{k}^{2}}n-m$$
$$\downarrow$$
$${{x}^{2}}-2kx-\frac{m}{n}+{{k}^{2}}$$
$$\downarrow$$
$$\left( x-\sqrt{\frac{m}{n}}-k\right) \,\left( x+\sqrt{\frac{m}{n}}-k\right) $$