Step 1 :

After factoring out $ 4 $ we have:

$$ 108x^{2}+120x-32 = 4 ( 27x^{2}+30x-8 ) $$

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:

$$ a = 27 , b = 30 ~ \text{ and } ~ c = -8 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 27 }$ by the constant term $\color{blue}{c = -8} $.

$$ a \cdot c = -216 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -216 $ and add to $ b = 30 $.

Step 5: All pairs of numbers with a product of $ -216 $ are:

PRODUCT = -216
-1     216	1     -216
-2     108	2     -108
-3     72	3     -72
-4     54	4     -54
-6     36	6     -36
-8     27	8     -27
-9     24	9     -24
-12     18	12     -18

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 30 }$

PRODUCT = -216 and SUM = 30
-1     216	1     -216
-2     108	2     -108
-3     72	3     -72
-4     54	4     -54
-6     36	6     -36
-8     27	8     -27
-9     24	9     -24
-12     18	12     -18

Step 7: Replace middle term $ 30 x $ with $ 36x-6x $:

$$ 27x^{2}+30x-8 = 27x^{2}+36x-6x-8 $$

Step 8: Apply factoring by grouping. Factor $ 9x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 27x^{2}+36x-6x-8 = 9x\left(3x+4\right) -2\left(3x+4\right) = \left(9x-2\right) \left(3x+4\right) $$

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