Factor polynomial $$ x^{2}+7x+10 $$

$$ x^{2}+7x+10 = ( x+2 )( x+5 ) $$

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:

$$ \color{blue}{ b = 7 } ~ \text{ and } ~ \color{red}{ c = 10 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 7 } $ and multiply to $ \color{red}{ 10 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 10 }$.

PRODUCT = 10
1     10	-1     -10
2     5	-2     -5

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

PRODUCT = 10 and SUM = 7
1     10	-1     -10
2     5	-2     -5

Step 4: Put 2 and 5 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+7x+10 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+7x+10 & = (x + 2)(x + 5) \end{aligned} $$

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