Factor polynomial $$ 3x^{2}-60x+225 $$

$$ 3x^{2}-60x+225 = 3( x-5 )( x-15 ) $$

Step 1 :

After factoring out $ 3 $ we have:

$$ 3x^{2}-60x+225 = 3 ( x^{2}-20x+75 ) $$

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:

$$ \color{blue}{ b = -20 } ~ \text{ and } ~ \color{red}{ c = 75 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -20 } $ and multiply to $ \color{red}{ 75 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 75 }$.

PRODUCT = 75
1     75	-1     -75
3     25	-3     -25
5     15	-5     -15

Step 4: Find out which pair sums up to $\color{blue}{ b = -20 }$

PRODUCT = 75 and SUM = -20
1     75	-1     -75
3     25	-3     -25
5     15	-5     -15

Step 5: Put -5 and -15 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-20x+75 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-20x+75 & = (x -5)(x -15) \end{aligned} $$

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