Factor polynomial $$ 6x^{2}-78x+216 $$

$$ 6x^{2}-78x+216 = 6( x-4 )( x-9 ) $$

Step 1 :

After factoring out $ 6 $ we have:

$$ 6x^{2}-78x+216 = 6 ( x^{2}-13x+36 ) $$

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:

$$ \color{blue}{ b = -13 } ~ \text{ and } ~ \color{red}{ c = 36 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -13 } $ and multiply to $ \color{red}{ 36 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 36 }$.

PRODUCT = 36
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 4: Find out which pair sums up to $\color{blue}{ b = -13 }$

PRODUCT = 36 and SUM = -13
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 5: Put -4 and -9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-13x+36 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-13x+36 & = (x -4)(x -9) \end{aligned} $$

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