Factor polynomial $$ 7x^{2}+56x+84 $$

$$ 7x^{2}+56x+84 = 7( x+2 )( x+6 ) $$

Step 1 :

After factoring out $ 7 $ we have:

$$ 7x^{2}+56x+84 = 7 ( x^{2}+8x+12 ) $$

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:

$$ \color{blue}{ b = 8 } ~ \text{ and } ~ \color{red}{ c = 12 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 8 } $ and multiply to $ \color{red}{ 12 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 12 }$.

PRODUCT = 12
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 4: Find out which pair sums up to $\color{blue}{ b = 8 }$

PRODUCT = 12 and SUM = 8
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 5: Put 2 and 6 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+8x+12 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+8x+12 & = (x + 2)(x + 6) \end{aligned} $$

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