Factor polynomial $$ 14m^{2}+15m-9 $$

$$ 14m^{2}+15m-9 = ( 7m-3 )( 2m+3 ) $$

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:

$$ a = 14 , b = 15 ~ \text{ and } ~ c = -9 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 14 }$ by the constant term $\color{blue}{c = -9} $.

$$ a \cdot c = -126 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -126 $ and add to $ b = 15 $.

Step 4: All pairs of numbers with a product of $ -126 $ are:

PRODUCT = -126
-1     126	1     -126
-2     63	2     -63
-3     42	3     -42
-6     21	6     -21
-7     18	7     -18
-9     14	9     -14

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 15 }$

PRODUCT = -126 and SUM = 15
-1     126	1     -126
-2     63	2     -63
-3     42	3     -42
-6     21	6     -21
-7     18	7     -18
-9     14	9     -14

Step 6: Replace middle term $ 15 x $ with $ 21x-6x $:

$$ 14x^{2}+15x-9 = 14x^{2}+21x-6x-9 $$

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

$$ 14x^{2}+21x-6x-9 = 7x\left(2x+3\right) -3\left(2x+3\right) = \left(7x-3\right) \left(2x+3\right) $$

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