Step 1 :

After factoring out $ 3 $ we have:

$$ 9x^{2}+66x+57 = 3 ( 3x^{2}+22x+19 ) $$

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:

$$ a = 3 , b = 22 ~ \text{ and } ~ c = 19 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 3 }$ by the constant term $\color{blue}{c = 19} $.

$$ a \cdot c = 57 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 57 $ and add to $ b = 22 $.

Step 5: All pairs of numbers with a product of $ 57 $ are:

PRODUCT = 57
1     57	-1     -57
3     19	-3     -19

Step 6: Find out which factor pair sums up to $\color{blue}{ b = 22 }$

PRODUCT = 57 and SUM = 22
1     57	-1     -57
3     19	-3     -19

Step 7: Replace middle term $ 22 x $ with $ 19x+3x $:

$$ 3x^{2}+22x+19 = 3x^{2}+19x+3x+19 $$

Step 8: Apply factoring by grouping. Factor $ x $ out of the first two terms and $ 1 $ out of the last two terms.

$$ 3x^{2}+19x+3x+19 = x\left(3x+19\right) + 1\left(3x+19\right) = \left(x+1\right) \left(3x+19\right) $$

Polynomial Factoring Calculator