Factor polynomial $$ x^{2}-8x-768 $$

$$ x^{2}-8x-768 = ( x+24 )( x-32 ) $$

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:

$$ \color{blue}{ b = -8 } ~ \text{ and } ~ \color{red}{ c = -768 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -8 } $ and multiply to $ \color{red}{ -768 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -768 }$.

PRODUCT = -768
-1     768	1     -768
-2     384	2     -384
-3     256	3     -256
-4     192	4     -192
-6     128	6     -128
-8     96	8     -96
-12     64	12     -64
-16     48	16     -48
-24     32	24     -32

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

PRODUCT = -768 and SUM = -8
-1     768	1     -768
-2     384	2     -384
-3     256	3     -256
-4     192	4     -192
-6     128	6     -128
-8     96	8     -96
-12     64	12     -64
-16     48	16     -48
-24     32	24     -32

Step 4: Put 24 and -32 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-8x-768 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-8x-768 & = (x + 24)(x -32) \end{aligned} $$

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