Factor polynomial $$ 12x^{3}-14x^{2}-30x $$

$$ 12x^{3}-14x^{2}-30x = 2x( 6x^{2}-7x-15 ) $$

Step 1 :

After factoring out $ 2x $ we have:

$$ 12x^{3}-14x^{2}-30x = 2x ( 6x^{2}-7x-15 ) $$

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:

$$ a = 6 , b = -7 ~ \text{ and } ~ c = -15 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -90 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -90 $ and add to $ b = -7 $.

Step 5: All pairs of numbers with a product of $ -90 $ are:

PRODUCT = -90
-1     90	1     -90
-2     45	2     -45
-3     30	3     -30
-5     18	5     -18
-6     15	6     -15
-9     10	9     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -7 }$

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

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