Factor polynomial $$ 2x^{3}+5x^{2}-12x $$

$$ 2x^{3}+5x^{2}-12x = x( 2x-3 )( x+4 ) $$

Step 1 :

After factoring out $ x $ we have:

$$ 2x^{3}+5x^{2}-12x = x ( 2x^{2}+5x-12 ) $$

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:

$$ a = 2 , b = 5 ~ \text{ and } ~ c = -12 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -12} $.

$$ a \cdot c = -24 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -24 $ and add to $ b = 5 $.

Step 5: All pairs of numbers with a product of $ -24 $ are:

PRODUCT = -24
-1     24	1     -24
-2     12	2     -12
-3     8	3     -8
-4     6	4     -6

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 5 }$

PRODUCT = -24 and SUM = 5
-1     24	1     -24
-2     12	2     -12
-3     8	3     -8
-4     6	4     -6

Step 7: Replace middle term $ 5 x $ with $ 8x-3x $:

$$ 2x^{2}+5x-12 = 2x^{2}+8x-3x-12 $$

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

$$ 2x^{2}+8x-3x-12 = 2x\left(x+4\right) -3\left(x+4\right) = \left(2x-3\right) \left(x+4\right) $$

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