Factor polynomial $$ x^{2}+13x+42 $$

$$ x^{2}+13x+42 = ( x+6 )( x+7 ) $$

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:

$$ \color{blue}{ b = 13 } ~ \text{ and } ~ \color{red}{ c = 42 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 13 } $ and multiply to $ \color{red}{ 42 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 42 }$.

PRODUCT = 42
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

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

PRODUCT = 42 and SUM = 13
1     42	-1     -42
2     21	-2     -21
3     14	-3     -14
6     7	-6     -7

Step 4: Put 6 and 7 into placeholders to get factored form.

$$ \begin{aligned} x^{2}+13x+42 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}+13x+42 & = (x + 6)(x + 7) \end{aligned} $$

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