Factor polynomial $$ x^{2}+18x+77 $$

$$ x^{2}+18x+77 = ( x+7 )( x+11 ) $$

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:

$$ \color{blue}{ b = 18 } ~ \text{ and } ~ \color{red}{ c = 77 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 18 } $ and multiply to $ \color{red}{ 77 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 77 }$.

PRODUCT = 77
1     77	-1     -77
7     11	-7     -11

Step 3: Find out which pair sums up to $\color{blue}{ b = 18 }$

PRODUCT = 77 and SUM = 18
1     77	-1     -77
7     11	-7     -11

Step 4: Put 7 and 11 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+18x+77 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+18x+77 & = (x + 7)(x + 11) \end{aligned} $$

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