Factor polynomial $$ 6x^{2}-14x-12 $$

$$ 6x^{2}-14x-12 = 2( x-3 )( 3x+2 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 6x^{2}-14x-12 = 2 ( 3x^{2}-7x-6 ) $$

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:

$$ a = 3 , b = -7 ~ \text{ and } ~ c = -6 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = -6} $.

$$ a \cdot c = -18 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -18 $ and add to $ b = -7 $.

Step 5: All pairs of numbers with a product of $ -18 $ are:

PRODUCT = -18
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -7 }$

PRODUCT = -18 and SUM = -7
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

Step 7: Replace middle term $ -7 x $ with $ 2x-9x $:

$$ 3x^{2}-7x-6 = 3x^{2}+2x-9x-6 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 3x^{2}+2x-9x-6 = x\left(3x+2\right) -3\left(3x+2\right) = \left(x-3\right) \left(3x+2\right) $$

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