Factor polynomial $$ 2w^{2}+3w-77 $$

$$ 2w^{2}+3w-77 = ( 2w-11 )( w+7 ) $$

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:

$$ a = 2 , b = 3 ~ \text{ and } ~ c = -77 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -77} $.

$$ a \cdot c = -154 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -154 $ and add to $ b = 3 $.

Step 4: All pairs of numbers with a product of $ -154 $ are:

PRODUCT = -154
-1     154	1     -154
-2     77	2     -77
-7     22	7     -22
-11     14	11     -14

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 3 }$

PRODUCT = -154 and SUM = 3
-1     154	1     -154
-2     77	2     -77
-7     22	7     -22
-11     14	11     -14

Step 6: Replace middle term $ 3 x $ with $ 14x-11x $:

$$ 2x^{2}+3x-77 = 2x^{2}+14x-11x-77 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -11 $ out of the last two terms.

$$ 2x^{2}+14x-11x-77 = 2x\left(x+7\right) -11\left(x+7\right) = \left(2x-11\right) \left(x+7\right) $$

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