Factor polynomial $$ 6x^{2}-7x-13 $$

$$ 6x^{2}-7x-13 = ( 6x-13 )( x+1 ) $$

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:

$$ a = 6 , b = -7 ~ \text{ and } ~ c = -13 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = -13} $.

$$ a \cdot c = -78 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -78 $ and add to $ b = -7 $.

Step 4: All pairs of numbers with a product of $ -78 $ are:

PRODUCT = -78
-1     78	1     -78
-2     39	2     -39
-3     26	3     -26
-6     13	6     -13

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -7 }$

PRODUCT = -78 and SUM = -7
-1     78	1     -78
-2     39	2     -39
-3     26	3     -26
-6     13	6     -13

Step 6: Replace middle term $ -7 x $ with $ 6x-13x $:

$$ 6x^{2}-7x-13 = 6x^{2}+6x-13x-13 $$

Step 7: Apply factoring by grouping. Factor $ 6x $ out of the first two terms and $ -13 $ out of the last two terms.

$$ 6x^{2}+6x-13x-13 = 6x\left(x+1\right) -13\left(x+1\right) = \left(6x-13\right) \left(x+1\right) $$

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