Factor polynomial $$ 2x^{2}+30x+108 $$

$$ 2x^{2}+30x+108 = 2( x+6 )( x+9 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 2x^{2}+30x+108 = 2 ( x^{2}+15x+54 ) $$

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:

$$ \color{blue}{ b = 15 } ~ \text{ and } ~ \color{red}{ c = 54 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 15 } $ and multiply to $ \color{red}{ 54 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 54 }$.

PRODUCT = 54
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 4: Find out which pair sums up to $\color{blue}{ b = 15 }$

PRODUCT = 54 and SUM = 15
1     54	-1     -54
2     27	-2     -27
3     18	-3     -18
6     9	-6     -9

Step 5: Put 6 and 9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+15x+54 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+15x+54 & = (x + 6)(x + 9) \end{aligned} $$

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