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