Factor polynomial $$ -2x^{3}-7x^{2}+15x $$

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

Step 1 :

After factoring out $ -x $ we have:

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

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:

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

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -30 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -30 $ and add to $ b = 7 $.

Step 5: 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 6: 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 7: Replace middle term $ 7 x $ with $ 10x-3x $:

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

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

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

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