Factor polynomial $$ c^{2}-3c-54 $$

$$ c^{2}-3c-54 = ( c+6 )( c-9 ) $$

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:

$$ \color{blue}{ b = -3 } ~ \text{ and } ~ \color{red}{ c = -54 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -3 } $ and multiply to $ \color{red}{ -54 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -54 }$.

PRODUCT = -54
-1     54	1     -54
-2     27	2     -27
-3     18	3     -18
-6     9	6     -9

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

PRODUCT = -54 and SUM = -3
-1     54	1     -54
-2     27	2     -27
-3     18	3     -18
-6     9	6     -9

Step 4: Put 6 and -9 into placeholders to get factored form.

$$ \begin{aligned} c^{2}-3c-54 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ c^{2}-3c-54 & = (x + 6)(x -9) \end{aligned} $$

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