Factor polynomial $$ x^{2}+13x+40 $$

$$ x^{2}+13x+40 = ( x+5 )( x+8 ) $$

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:

$$ \color{blue}{ b = 13 } ~ \text{ and } ~ \color{red}{ c = 40 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 13 } $ and multiply to $ \color{red}{ 40 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 40 }$.

PRODUCT = 40
1     40	-1     -40
2     20	-2     -20
4     10	-4     -10
5     8	-5     -8

Step 3: Find out which pair sums up to $\color{blue}{ b = 13 }$

PRODUCT = 40 and SUM = 13
1     40	-1     -40
2     20	-2     -20
4     10	-4     -10
5     8	-5     -8

Step 4: Put 5 and 8 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+13x+40 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+13x+40 & = (x + 5)(x + 8) \end{aligned} $$

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