Step 1 :

After factoring out $ -3x $ we have:

$$ -3x^{3}+45x^{2}-135x = -3x ( x^{2}-15x+45 ) $$

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:

$$ \color{blue}{ b = -15 } ~ \text{ and } ~ \color{red}{ c = 45 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -15 } $ 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: Because none of these pairs will give us a sum of $ \color{blue}{ -15 }$, we conclude the polynomial cannot be factored.

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