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. 

Assuming your question is in earnest and you aren't trolling the answer is
$ax^2 + bx + c = a(x  + \frac{b + \sqrt{b^2 - 4ac}}{2a})(x  + \frac{b - \sqrt{b^2 - 4ac}}{2a} )$
No guessing.  If the equation factors at all (which it won't if $4ac > b^2$) then it will always factor to that.
