Factor polynomial $$ x^{3}+9x^{2}-52x $$

$$ x^{3}+9x^{2}-52x = x( x-4 )( x+13 ) $$

Step 1 :

After factoring out $ x $ we have:

$$ x^{3}+9x^{2}-52x = x ( x^{2}+9x-52 ) $$

Step 2 :

Step 2: 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:

$$ \color{blue}{ b = 9 } ~ \text{ and } ~ \color{red}{ c = -52 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 9 } $ and multiply to $ \color{red}{ -52 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = -52 }$.

PRODUCT = -52
-1     52	1     -52
-2     26	2     -26
-4     13	4     -13

Step 4: Find out which pair sums up to $\color{blue}{ b = 9 }$

PRODUCT = -52 and SUM = 9
-1     52	1     -52
-2     26	2     -26
-4     13	4     -13

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

$$ \begin{aligned} x^{2}+9x-52 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+9x-52 & = (x -4)(x + 13) \end{aligned} $$

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