Factor polynomial $$ x^{2}+10x-75 $$

$$ x^{2}+10x-75 = ( x-5 )( x+15 ) $$

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:

$$ \color{blue}{ b = 10 } ~ \text{ and } ~ \color{red}{ c = -75 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 10 } $ and multiply to $ \color{red}{ -75 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -75 }$.

PRODUCT = -75
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 3: Find out which pair sums up to $\color{blue}{ b = 10 }$

PRODUCT = -75 and SUM = 10
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 4: Put -5 and 15 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+10x-75 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+10x-75 & = (x -5)(x + 15) \end{aligned} $$

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