2x²-5x-12 = 0

We must factor the left side:

Multiply the 2 by the 12 ignoring signs.  Get 24

Write down all the ways to have two positive integers
which have product 24, starting with 24*1

24*1
12*2
 8*3
 6*4

Since the last sign in 2x²-5x-12 is -, SUBTRACT them,
and place the DIFFERENCE out beside that:

24*1   24-1=23
12*2   12-2=10
 8*3    8-3=5
 6*4    6-4=2

Now, again ignoring signs, we find in that list of
differences the coefficient of the middle term in 2x²-5x-12

So we replace the number 5 by 8-3

2x²-5x-3 = 0
2x²-(8-3)x-12 = 0

Then we distribute to remove the parentheses:

2x²-8x+3x-12 = 0

Factor the first two terms 2x²-8 by taking out the
greatest common factor, 2x, getting 2x(x-4)

Factor the last two terms +3x-12 by taking out the
greatest common factor, getting +3(x-4)

So we have

2x(x-4)+3(x-4) = 0

Notice that there is a common factor, (x-4)

2x(x-4)+3(x-4) = 0

which we can factor out leaving the 2x and the +3 to put 
in parentheses:

(x-4)(2x+3) = 0

Now we use the zero-factor property:

x-4 = 0,  2x+3 = 0
  x = 4,    2x = -3
             x = 



---------------------------

Here's another example where the last term is + :

12x²-32x+21 = 0

Multiply the 12 by the 21 ignoring signs.  Get 252

Write down all the ways to have two positive integers
which have product 252, starting with 252*1

252*1
126*2
 84*3
 63*4
 42*6
 36*7
 28*9
 21*12
 18*14

Since the last sign in 12x²-32x+21 is +, ADD them,
and place the SUM out beside that:


252*1    252+1=253
126*2    126+2=128
 84*3     84+3-87
 63*4     63+4=67
 42*6     42+6=48
 36*7     36+7=43
 28*9     28+9=37 
 21*12   21+12=33
 18*14   18+14=32


Now, again ignoring signs, we find in that list of
sums the coefficient of the middle term in 12x²-32x+21

So we replace the number 32 by 18+14

12x²-32x+21 = 0
12x²-(18+14)x+21 = 0

Then we distribute to remove the parentheses:

12x²-18x-14x+21 = 0

Factor the first two terms 12x²-18x by taking out the
greatest common factor, getting 6x(2x-3)

Factor the last two terms -14x+21 by taking out the
greatest common factor, -7, getting -7(2x-3)

So we have

6x(2x-3)-7(2x-3) = 0

Notice that there is a common factor, (2x-3)

6x(2x-3)-7(2x-3) = 0

which we can factor out leaving the 6x and the -7 to put 
in parentheses:

(2x-3)(6x-7) = 0

Using the zero factor property:

    2x-3 = 0          6x-7 = 0
      2x = 3            6x = 7  
       x =              x = 

Edwin