Factor polynomial $$ 14x^{2}+91x+77 $$

$$ 14x^{2}+91x+77 = 7( x+1 )( 2x+11 ) $$

Step 1 :

After factoring out $ 7 $ we have:

$$ 14x^{2}+91x+77 = 7 ( 2x^{2}+13x+11 ) $$

Step 2 :

Step 2: 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:

$$ a = 2 , b = 13 ~ \text{ and } ~ c = 11 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 11} $.

$$ a \cdot c = 22 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 22 $ and add to $ b = 13 $.

Step 5: All pairs of numbers with a product of $ 22 $ are:

PRODUCT = 22
1     22	-1     -22
2     11	-2     -11

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 13 }$

PRODUCT = 22 and SUM = 13
1     22	-1     -22
2     11	-2     -11

Step 7: Replace middle term $ 13 x $ with $ 11x+2x $:

$$ 2x^{2}+13x+11 = 2x^{2}+11x+2x+11 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 2x^{2}+11x+2x+11 = x\left(2x+11\right) + 1\left(2x+11\right) = \left(x+1\right) \left(2x+11\right) $$

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