Factor polynomial $$ 2x^{2}-25x-247 $$

$$ 2x^{2}-25x-247 = ( x-19 )( 2x+13 ) $$

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:

$$ a = 2 , b = -25 ~ \text{ and } ~ c = -247 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -247} $.

$$ a \cdot c = -494 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -494 $ and add to $ b = -25 $.

Step 4: All pairs of numbers with a product of $ -494 $ are:

PRODUCT = -494
-1     494	1     -494
-2     247	2     -247
-13     38	13     -38
-19     26	19     -26

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -25 }$

PRODUCT = -494 and SUM = -25
-1     494	1     -494
-2     247	2     -247
-13     38	13     -38
-19     26	19     -26

Step 6: Replace middle term $ -25 x $ with $ 13x-38x $:

$$ 2x^{2}-25x-247 = 2x^{2}+13x-38x-247 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -19 $ out of the last two terms.

$$ 2x^{2}+13x-38x-247 = x\left(2x+13\right) -19\left(2x+13\right) = \left(x-19\right) \left(2x+13\right) $$

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