Step 1 :

After factoring out $ 3 $ we have:

$$ 12k^{2}-111k+27 = 3 ( 4k^{2}-37k+9 ) $$

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:

$$ a = 4 , b = -37 ~ \text{ and } ~ c = 9 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = 9} $.

$$ a \cdot c = 36 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 36 $ and add to $ b = -37 $.

Step 5: All pairs of numbers with a product of $ 36 $ are:

PRODUCT = 36
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -37 }$

PRODUCT = 36 and SUM = -37
1     36	-1     -36
2     18	-2     -18
3     12	-3     -12
4     9	-4     -9
6     6	-6     -6

Step 7: Replace middle term $ -37 x $ with $ -x-36x $:

$$ 4x^{2}-37x+9 = 4x^{2}-x-36x+9 $$

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

$$ 4x^{2}-x-36x+9 = x\left(4x-1\right) -9\left(4x-1\right) = \left(x-9\right) \left(4x-1\right) $$

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