Factor polynomial $$ n^{2}+10n+21 $$

$$ n^{2}+10n+21 = ( n+3 )( n+7 ) $$

Step 1: 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:

$$ \color{blue}{ b = 10 } ~ \text{ and } ~ \color{red}{ c = 21 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 10 } $ and multiply to $ \color{red}{ 21 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 21 }$.

PRODUCT = 21
1     21	-1     -21
3     7	-3     -7

Step 3: Find out which pair sums up to $\color{blue}{ b = 10 }$

PRODUCT = 21 and SUM = 10
1     21	-1     -21
3     7	-3     -7

Step 4: Put 3 and 7 into placeholders to get factored form.

$$ \begin{aligned} n^{2}+10n+21 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ n^{2}+10n+21 & = (x + 3)(x + 7) \end{aligned} $$

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