Different ways to factor 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. 

Variation of the method mentioned by @suomnonA
1) Multiply $a*c$.
2) Find a factor pair of $ac$ that adds up to $b$. Let them be $x_1$ and $x_2$.
3) Write factors as:
$$\frac{(ax-x_1)(ax-x_2)}{a}$$
The factors on top will factorize down to cancel out with the $a$ on the bottom.
Example:
$$6x^2+17x-14$$
$$a=6, b=17, c=-14$$
Here we do step $1$, getting $$ac=-84$$ Then we do step $2$. The factor pair is $$-4,21$$ Now we do step $3$. 
The equation becomes $$\frac{(6x-4)(6x+21)}{2}$$
which factorizes to
$$\frac{2(3x-2)\cdot3(2x+7)}{6}$$
$$(3x-2)(2x+7)$$
