Factor polynomial $$ n^{2}-8n-65 $$

$$ n^{2}-8n-65 = ( n+5 )( n-13 ) $$

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:

$$ \color{blue}{ b = -8 } ~ \text{ and } ~ \color{red}{ c = -65 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -8 } $ and multiply to $ \color{red}{ -65 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -65 }$.

PRODUCT = -65
-1     65	1     -65
-5     13	5     -13

Step 3: Find out which pair sums up to $\color{blue}{ b = -8 }$

PRODUCT = -65 and SUM = -8
-1     65	1     -65
-5     13	5     -13

Step 4: Put 5 and -13 into placeholders to get factored form.

$$ \begin{aligned} n^{2}-8n-65 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ n^{2}-8n-65 & = (x + 5)(x -13) \end{aligned} $$

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