Ax^2 + Bx + C = 0  can be factored using its coefficients.

Consider representing factors as fractions;

(3x-2)  ->  -2/3

(4x+5)  ->  5/4

(6x-7)   ->  -7/6

(x - 3)   ->  -3/1

You can factor the quadratic

12x^2   - 5x - 2

by using the coefficients A, B and C to form fractions, that represent the factors.

A= 12

B =  -5
C = -2

How to form the fractions using A, B, C:
First multiply AC (12)(-2) = -24

Now find the factors of AC (-24) that add to B (-5).
Factors of AC (-24) that add to B (-5) are 3 and -8  (  -24 =  -8*3 )

Form two fractions using -8 and 3 as the numerator.
The denominator of each fraction is A which is 12;  reduce each fractions

-8/12 and 3/12   reduce to -2/3   and 1/4.  These latter two fractions represent the factors.

-2/3 - > (3x-2)
1/4   -> (4x+1)

Hence 12x^2   - 5x - 2 = (3x-2)(4x+1)

QED.