Step 1 :

After factoring out $ -1 $ we have:

$$ -x^{2}+19x-34 = - ~ ( x^{2}-19x+34 ) $$

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:

$$ \color{blue}{ b = -19 } ~ \text{ and } ~ \color{red}{ c = 34 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -19 } $ and multiply to $ \color{red}{ 34 } $.

Step 3: Find out pairs of numbers with a product of $\color{red}{ c = 34 }$.

PRODUCT = 34
1     34	-1     -34
2     17	-2     -17

Step 4: Find out which pair sums up to $\color{blue}{ b = -19 }$

PRODUCT = 34 and SUM = -19
1     34	-1     -34
2     17	-2     -17

Step 5: Put -2 and -17 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-19x+34 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-19x+34 & = (x -2)(x -17) \end{aligned} $$

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