Step 1 :

After factoring out $ 3x^{2} $ we have:

$$ 9x^{4}+3x^{3}+15x^{2} = 3x^{2} ( 3x^{2}+x+5 ) $$

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:

$$ a = 3 , b = 1 ~ \text{ and } ~ c = 5 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 5} $.

$$ a \cdot c = 15 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 15 $ and add to $ b = 1 $.

Step 5: All pairs of numbers with a product of $ 15 $ are:

PRODUCT = 15
1     15	-1     -15
3     5	-3     -5

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 1 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ 1 }$, we conclude the polynomial cannot be factored.

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