Factor polynomial $$ -2x^{3}+3x^{2}+48x $$

$$ -2x^{3}+3x^{2}+48x = -x( 2x^{2}-3x-48 ) $$

Step 1 :

After factoring out $ -x $ we have:

$$ -2x^{3}+3x^{2}+48x = -x ( 2x^{2}-3x-48 ) $$

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:

$$ a = 2 , b = -3 ~ \text{ and } ~ c = -48 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -48} $.

$$ a \cdot c = -96 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -96 $ and add to $ b = -3 $.

Step 5: All pairs of numbers with a product of $ -96 $ are:

PRODUCT = -96
-1     96	1     -96
-2     48	2     -48
-3     32	3     -32
-4     24	4     -24
-6     16	6     -16
-8     12	8     -12

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -3 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -3 }$, we conclude the polynomial cannot be factored.

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