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:

$$ \color{blue}{ b = -5 } ~ \text{ and } ~ \color{red}{ c = -84 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -5 } $ and multiply to $ \color{red}{ -84 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -84 }$.

PRODUCT = -84
-1     84	1     -84
-2     42	2     -42
-3     28	3     -28
-4     21	4     -21
-6     14	6     -14
-7     12	7     -12

Step 3: Find out which pair sums up to $\color{blue}{ b = -5 }$

PRODUCT = -84 and SUM = -5
-1     84	1     -84
-2     42	2     -42
-3     28	3     -28
-4     21	4     -21
-6     14	6     -14
-7     12	7     -12

Step 4: Put 7 and -12 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-5x-84 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-5x-84 & = (x + 7)(x -12) \end{aligned} $$

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