Factor polynomial $$ z^{2}+7z+12 $$

$$ z^{2}+7z+12 = ( z+3 )( z+4 ) $$

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:

$$ \color{blue}{ b = 7 } ~ \text{ and } ~ \color{red}{ c = 12 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ 7 } $ and multiply to $ \color{red}{ 12 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 12 }$.

PRODUCT = 12
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

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

PRODUCT = 12 and SUM = 7
1     12	-1     -12
2     6	-2     -6
3     4	-3     -4

Step 4: Put 3 and 4 into placeholders to get factored form.

$$ \begin{aligned} z^{2}+7z+12 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ z^{2}+7z+12 & = (x + 3)(x + 4) \end{aligned} $$

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