Factor polynomial $$ 7x^{2}-8x-12 $$

$$ 7x^{2}-8x-12 = ( x-2 )( 7x+6 ) $$

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:

$$ a = 7 , b = -8 ~ \text{ and } ~ c = -12 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -12} $.

$$ a \cdot c = -84 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -84 $ and add to $ b = -8 $.

Step 4: All pairs of numbers with a product of $ -84 $ are:

PRODUCT = -84
-1     84	1     -84
-2     42	2     -42
-3     28	3     -28
-4     21	4     -21
-6     14	6     -14
-7     12	7     -12

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -8 }$

PRODUCT = -84 and SUM = -8
-1     84	1     -84
-2     42	2     -42
-3     28	3     -28
-4     21	4     -21
-6     14	6     -14
-7     12	7     -12

Step 6: Replace middle term $ -8 x $ with $ 6x-14x $:

$$ 7x^{2}-8x-12 = 7x^{2}+6x-14x-12 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 7x^{2}+6x-14x-12 = x\left(7x+6\right) -2\left(7x+6\right) = \left(x-2\right) \left(7x+6\right) $$

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