Factor polynomial $$ x^{2}-2x-63 $$

$$ x^{2}-2x-63 = ( x+7 )( x-9 ) $$

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:

$$ \color{blue}{ b = -2 } ~ \text{ and } ~ \color{red}{ c = -63 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -2 } $ and multiply to $ \color{red}{ -63 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = -63 }$.

PRODUCT = -63
-1     63	1     -63
-3     21	3     -21
-7     9	7     -9

Step 3: Find out which pair sums up to $\color{blue}{ b = -2 }$

PRODUCT = -63 and SUM = -2
-1     63	1     -63
-3     21	3     -21
-7     9	7     -9

Step 4: Put 7 and -9 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-2x-63 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-2x-63 & = (x + 7)(x -9) \end{aligned} $$

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