Factor polynomial $$ 2x^{2}-7x-15 $$

$$ 2x^{2}-7x-15 = ( x-5 )( 2x+3 ) $$

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:

$$ a = 2 , b = -7 ~ \text{ and } ~ c = -15 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -30 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -30 $ and add to $ b = -7 $.

Step 4: All pairs of numbers with a product of $ -30 $ are:

PRODUCT = -30
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

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

PRODUCT = -30 and SUM = -7
-1     30	1     -30
-2     15	2     -15
-3     10	3     -10
-5     6	5     -6

Step 6: Replace middle term $ -7 x $ with $ 3x-10x $:

$$ 2x^{2}-7x-15 = 2x^{2}+3x-10x-15 $$

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

$$ 2x^{2}+3x-10x-15 = x\left(2x+3\right) -5\left(2x+3\right) = \left(x-5\right) \left(2x+3\right) $$

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