Step 1: 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:

$$ a = 34 , b = 1 ~ \text{ and } ~ c = -8 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 34 }$ by the constant term $\color{blue}{c = -8} $.

$$ a \cdot c = -272 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -272 $ and add to $ b = 1 $.

Step 4: All pairs of numbers with a product of $ -272 $ are:

PRODUCT = -272
-1     272	1     -272
-2     136	2     -136
-4     68	4     -68
-8     34	8     -34
-16     17	16     -17

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 1 }$

PRODUCT = -272 and SUM = 1
-1     272	1     -272
-2     136	2     -136
-4     68	4     -68
-8     34	8     -34
-16     17	16     -17

Step 6: Replace middle term $ 1 x $ with $ 17x-16x $:

$$ 34x^{2}+x-8 = 34x^{2}+17x-16x-8 $$

Step 7: Apply factoring by grouping. Factor $ 17x $ out of the first two terms and $ -8 $ out of the last two terms.

$$ 34x^{2}+17x-16x-8 = 17x\left(2x+1\right) -8\left(2x+1\right) = \left(17x-8\right) \left(2x+1\right) $$

Polynomial Factoring Calculator