Factor polynomial $$ x^{2}+7x-78 $$

$$ x^{2}+7x-78 = ( x-6 )( x+13 ) $$

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:

$$ \color{blue}{ b = 7 } ~ \text{ and } ~ \color{red}{ c = -78 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 7 } $ and multiply to $ \color{red}{ -78 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -78 }$.

PRODUCT = -78
-1     78	1     -78
-2     39	2     -39
-3     26	3     -26
-6     13	6     -13

Step 3: Find out which pair sums up to $\color{blue}{ b = 7 }$

PRODUCT = -78 and SUM = 7
-1     78	1     -78
-2     39	2     -39
-3     26	3     -26
-6     13	6     -13

Step 4: Put -6 and 13 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+7x-78 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+7x-78 & = (x -6)(x + 13) \end{aligned} $$

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