Factor polynomial $$ x^{2}-14x-51 $$

$$ x^{2}-14x-51 = ( x+3 )( x-17 ) $$

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:

$$ \color{blue}{ b = -14 } ~ \text{ and } ~ \color{red}{ c = -51 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -14 } $ and multiply to $ \color{red}{ -51 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -51 }$.

PRODUCT = -51
-1     51	1     -51
-3     17	3     -17

Step 3: Find out which pair sums up to $\color{blue}{ b = -14 }$

PRODUCT = -51 and SUM = -14
-1     51	1     -51
-3     17	3     -17

Step 4: Put 3 and -17 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-14x-51 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-14x-51 & = (x + 3)(x -17) \end{aligned} $$

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