Step 1 :

After factoring out $ 6 $ we have:

$$ 6x^{2}-42x+54 = 6 ( x^{2}-7x+9 ) $$

Step 2 :

Step 2: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = -7 } ~ \text{ and } ~ \color{red}{ c = 9 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -7 } $ and multiply to $ \color{red}{ 9 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 9 }$.

PRODUCT = 9
1     9	-1     -9
3     3	-3     -3

Step 4: 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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