Factor polynomial $$ 12x^{2}-8x-15 $$

$$ 12x^{2}-8x-15 = ( 2x-3 )( 6x+5 ) $$

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:

$$ a = 12 , b = -8 ~ \text{ and } ~ c = -15 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 12 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -180 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -180 $ and add to $ b = -8 $.

Step 4: All pairs of numbers with a product of $ -180 $ are:

PRODUCT = -180
-1     180	1     -180
-2     90	2     -90
-3     60	3     -60
-4     45	4     -45
-5     36	5     -36
-6     30	6     -30
-9     20	9     -20
-10     18	10     -18
-12     15	12     -15

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

PRODUCT = -180 and SUM = -8
-1     180	1     -180
-2     90	2     -90
-3     60	3     -60
-4     45	4     -45
-5     36	5     -36
-6     30	6     -30
-9     20	9     -20
-10     18	10     -18
-12     15	12     -15

Step 6: Replace middle term $ -8 x $ with $ 10x-18x $:

$$ 12x^{2}-8x-15 = 12x^{2}+10x-18x-15 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 12x^{2}+10x-18x-15 = 2x\left(6x+5\right) -3\left(6x+5\right) = \left(2x-3\right) \left(6x+5\right) $$

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