Factor polynomial $$ -3x^{3}+27x^{2}-60x $$

$$ -3x^{3}+27x^{2}-60x = ( -3 )x( x-4 )( x-5 ) $$

Step 1 :

After factoring out $ -3x $ we have:

$$ -3x^{3}+27x^{2}-60x = -3x ( x^{2}-9x+20 ) $$

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:

$$ \color{blue}{ b = -9 } ~ \text{ and } ~ \color{red}{ c = 20 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -9 } $ and multiply to $ \color{red}{ 20 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 20 }$.

PRODUCT = 20
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 4: Find out which pair sums up to $\color{blue}{ b = -9 }$

PRODUCT = 20 and SUM = -9
1     20	-1     -20
2     10	-2     -10
4     5	-4     -5

Step 5: Put -4 and -5 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-9x+20 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-9x+20 & = (x -4)(x -5) \end{aligned} $$

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