Factor polynomial $$ 6m^{2}+5m-21 $$

$$ 6m^{2}+5m-21 = ( 2m-3 )( 3m+7 ) $$

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:

$$ a = 6 , b = 5 ~ \text{ and } ~ c = -21 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = -21} $.

$$ a \cdot c = -126 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -126 $ and add to $ b = 5 $.

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 $\color{blue}{ b = 5 }$

PRODUCT = -126 and SUM = 5
-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 $ 5 x $ with $ 14x-9x $:

$$ 6x^{2}+5x-21 = 6x^{2}+14x-9x-21 $$

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

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

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