Factor polynomial $$ 12x^{2}-143x-12 $$

$$ 12x^{2}-143x-12 = ( x-12 )( 12x+1 ) $$

Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 12 , b = -143 ~ \text{ and } ~ c = -12 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -12} $.

$$ a \cdot c = -144 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -144 $ and add to $ b = -143 $.

Step 4: All pairs of numbers with a product of $ -144 $ are:

PRODUCT = -144
-1     144	1     -144
-2     72	2     -72
-3     48	3     -48
-4     36	4     -36
-6     24	6     -24
-8     18	8     -18
-9     16	9     -16
-12     12	12     -12

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -143 }$

PRODUCT = -144 and SUM = -143
-1     144	1     -144
-2     72	2     -72
-3     48	3     -48
-4     36	4     -36
-6     24	6     -24
-8     18	8     -18
-9     16	9     -16
-12     12	12     -12

Step 6: Replace middle term $ -143 x $ with $ x-144x $:

$$ 12x^{2}-143x-12 = 12x^{2}+x-144x-12 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -12 $ out of the last two terms.

$$ 12x^{2}+x-144x-12 = x\left(12x+1\right) -12\left(12x+1\right) = \left(x-12\right) \left(12x+1\right) $$

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