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:

$$ \color{blue}{ b = 14 } ~ \text{ and } ~ \color{red}{ c = 48 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 14 } $ and multiply to $ \color{red}{ 48 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 48 }$.

PRODUCT = 48
1     48	-1     -48
2     24	-2     -24
3     16	-3     -16
4     12	-4     -12
6     8	-6     -8

Step 3: Find out which pair sums up to $\color{blue}{ b = 14 }$

PRODUCT = 48 and SUM = 14
1     48	-1     -48
2     24	-2     -24
3     16	-3     -16
4     12	-4     -12
6     8	-6     -8

Step 4: Put 6 and 8 into placeholders to get factored form.

$$ \begin{aligned} n^{2}+14n+48 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ n^{2}+14n+48 & = (x + 6)(x + 8) \end{aligned} $$

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