Factor polynomial $$ 9p^{2}-9p+2 $$

$$ 9p^{2}-9p+2 = ( 3p-2 )( 3p-1 ) $$

Step 1: 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:

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

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

$$ a \cdot c = 18 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 18 $ and add to $ b = -9 $.

Step 4: 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 5: Find out which factor pair sums up to $\color{blue}{ b = -9 }$

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

Step 6: Replace middle term $ -9 x $ with $ -3x-6x $:

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

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -2 $ out of the last two terms.

$$ 9x^{2}-3x-6x+2 = 3x\left(3x-1\right) -2\left(3x-1\right) = \left(3x-2\right) \left(3x-1\right) $$

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