Step 1 :

After factoring out $ 3 $ we have:

$$ 6a^{2}+9a-27 = 3 ( 2a^{2}+3a-9 ) $$

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:

$$ a = 2 , b = 3 ~ \text{ and } ~ c = -9 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = -9} $.

$$ a \cdot c = -18 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -18 $ and add to $ b = 3 $.

Step 5: All pairs of numbers with a product of $ -18 $ are:

PRODUCT = -18
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

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

PRODUCT = -18 and SUM = 3
-1     18	1     -18
-2     9	2     -9
-3     6	3     -6

Step 7: Replace middle term $ 3 x $ with $ 6x-3x $:

$$ 2x^{2}+3x-9 = 2x^{2}+6x-3x-9 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

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

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