Factor polynomial $$ 2x^{2}-5x-25 $$

$$ 2x^{2}-5x-25 = ( x-5 )( 2x+5 ) $$

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:

$$ a = 2 , b = -5 ~ \text{ and } ~ c = -25 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -25} $.

$$ a \cdot c = -50 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -50 $ and add to $ b = -5 $.

Step 4: All pairs of numbers with a product of $ -50 $ are:

PRODUCT = -50
-1     50	1     -50
-2     25	2     -25
-5     10	5     -10

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

PRODUCT = -50 and SUM = -5
-1     50	1     -50
-2     25	2     -25
-5     10	5     -10

Step 6: Replace middle term $ -5 x $ with $ 5x-10x $:

$$ 2x^{2}-5x-25 = 2x^{2}+5x-10x-25 $$

Step 7: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 2x^{2}+5x-10x-25 = x\left(2x+5\right) -5\left(2x+5\right) = \left(x-5\right) \left(2x+5\right) $$

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