Step 1 :

After factoring out $ 3 $ we have:

$$ 9x^{2}-48x+54 = 3 ( 3x^{2}-16x+18 ) $$

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:

$$ a = 3 , b = -16 ~ \text{ and } ~ c = 18 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 18} $.

$$ a \cdot c = 54 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 54 $ and add to $ b = -16 $.

Step 5: All pairs of numbers with a product of $ 54 $ are:

PRODUCT = 54
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -16 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -16 }$, we conclude the polynomial cannot be factored.

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