Factor polynomial $$ 12x^{2}-6x-168 $$

$$ 12x^{2}-6x-168 = 6( x-4 )( 2x+7 ) $$

Step 1 :

After factoring out $ 6 $ we have:

$$ 12x^{2}-6x-168 = 6 ( 2x^{2}-x-28 ) $$

Step 2 :

Step 2: 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:

$$ a = 2 , b = -1 ~ \text{ and } ~ c = -28 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -28} $.

$$ a \cdot c = -56 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -56 $ and add to $ b = -1 $.

Step 5: All pairs of numbers with a product of $ -56 $ are:

PRODUCT = -56
-1     56	1     -56
-2     28	2     -28
-4     14	4     -14
-7     8	7     -8

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -1 }$

PRODUCT = -56 and SUM = -1
-1     56	1     -56
-2     28	2     -28
-4     14	4     -14
-7     8	7     -8

Step 7: Replace middle term $ -1 x $ with $ 7x-8x $:

$$ 2x^{2}-x-28 = 2x^{2}+7x-8x-28 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -4 $ out of the last two terms.

$$ 2x^{2}+7x-8x-28 = x\left(2x+7\right) -4\left(2x+7\right) = \left(x-4\right) \left(2x+7\right) $$

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