Step 1 :

After factoring out $ 3 $ we have:

$$ 90n^{2}-51n+6 = 3 ( 30n^{2}-17n+2 ) $$

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:

$$ a = 30 , b = -17 ~ \text{ and } ~ c = 2 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 30 }$ by the constant term $\color{blue}{c = 2} $.

$$ a \cdot c = 60 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 60 $ and add to $ b = -17 $.

Step 5: All pairs of numbers with a product of $ 60 $ are:

PRODUCT = 60
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -17 }$

PRODUCT = 60 and SUM = -17
1     60	-1     -60
2     30	-2     -30
3     20	-3     -20
4     15	-4     -15
5     12	-5     -12
6     10	-6     -10

Step 7: Replace middle term $ -17 x $ with $ -5x-12x $:

$$ 30x^{2}-17x+2 = 30x^{2}-5x-12x+2 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 30x^{2}-5x-12x+2 = 5x\left(6x-1\right) -2\left(6x-1\right) = \left(5x-2\right) \left(6x-1\right) $$

Polynomial Factoring Calculator