Step 1 :

After factoring out $ 2a $ we have:

$$ 12a^{3}+6a^{2}+8a = 2a ( 6a^{2}+3a+4 ) $$

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:

$$ a = 6 , b = 3 ~ \text{ and } ~ c = 4 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 24 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 24 $ and add to $ b = 3 $.

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 $\color{blue}{ b = 3 }$

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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