Step 1 :

After factoring out $ -1 $ we have:

$$ -x^{2}+22x-72 = - ~ ( x^{2}-22x+72 ) $$

Step 2 :

Step 2: 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:

$$ \color{blue}{ b = -22 } ~ \text{ and } ~ \color{red}{ c = 72 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -22 } $ and multiply to $ \color{red}{ 72 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 72 }$.

PRODUCT = 72
1     72	-1     -72
2     36	-2     -36
3     24	-3     -24
4     18	-4     -18
6     12	-6     -12
8     9	-8     -9

Step 4: Find out which pair sums up to $\color{blue}{ b = -22 }$

PRODUCT = 72 and SUM = -22
1     72	-1     -72
2     36	-2     -36
3     24	-3     -24
4     18	-4     -18
6     12	-6     -12
8     9	-8     -9

Step 5: Put -4 and -18 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-22x+72 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-22x+72 & = (x -4)(x -18) \end{aligned} $$

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