Step 1 :

After factoring out $ -1 $ we have:

$$ -5x^{2}+3x+34 = - ~ ( 5x^{2}-3x-34 ) $$

Step 2 :

Step 2: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:

$$ a = 5 , b = -3 ~ \text{ and } ~ c = -34 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = -34} $.

$$ a \cdot c = -170 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -170 $ and add to $ b = -3 $.

Step 5: All pairs of numbers with a product of $ -170 $ are:

PRODUCT = -170
-1     170	1     -170
-2     85	2     -85
-5     34	5     -34
-10     17	10     -17

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -3 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -3 }$, we conclude the polynomial cannot be factored.

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