Step 1 :

After factoring out $ 3 $ we have:

$$ 3x^{2}-42x+135 = 3 ( x^{2}-14x+45 ) $$

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:

$$ \color{blue}{ b = -14 } ~ \text{ and } ~ \color{red}{ c = 45 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -14 } $ and multiply to $ \color{red}{ 45 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 45 }$.

PRODUCT = 45
1     45	-1     -45
3     15	-3     -15
5     9	-5     -9

Step 4: Find out which pair sums up to $\color{blue}{ b = -14 }$

PRODUCT = 45 and SUM = -14
1     45	-1     -45
3     15	-3     -15
5     9	-5     -9

Step 5: Put -5 and -9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-14x+45 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-14x+45 & = (x -5)(x -9) \end{aligned} $$

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