Factor polynomial $$ 12x^{2}-276x+1080 $$

$$ 12x^{2}-276x+1080 = 12( x-5 )( x-18 ) $$

Step 1 :

After factoring out $ 12 $ we have:

$$ 12x^{2}-276x+1080 = 12 ( x^{2}-23x+90 ) $$

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:

$$ \color{blue}{ b = -23 } ~ \text{ and } ~ \color{red}{ c = 90 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -23 } $ and multiply to $ \color{red}{ 90 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 90 }$.

PRODUCT = 90
1     90	-1     -90
2     45	-2     -45
3     30	-3     -30
5     18	-5     -18
6     15	-6     -15
9     10	-9     -10

Step 4: Find out which pair sums up to $\color{blue}{ b = -23 }$

PRODUCT = 90 and SUM = -23
1     90	-1     -90
2     45	-2     -45
3     30	-3     -30
5     18	-5     -18
6     15	-6     -15
9     10	-9     -10

Step 5: Put -5 and -18 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-23x+90 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-23x+90 & = (x -5)(x -18) \end{aligned} $$

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