Factor polynomial $$ 2n^{2}-5n-42 $$

$$ 2n^{2}-5n-42 = ( n-6 )( 2n+7 ) $$

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:

$$ a = 2 , b = -5 ~ \text{ and } ~ c = -42 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -42} $.

$$ a \cdot c = -84 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -84 $ and add to $ b = -5 $.

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

PRODUCT = -84 and SUM = -5
-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 $ -5 x $ with $ 7x-12x $:

$$ 2x^{2}-5x-42 = 2x^{2}+7x-12x-42 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -6 $ out of the last two terms.

$$ 2x^{2}+7x-12x-42 = x\left(2x+7\right) -6\left(2x+7\right) = \left(x-6\right) \left(2x+7\right) $$

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