Factor polynomial $$ 4x^{2}-18x+18 $$

$$ 4x^{2}-18x+18 = 2( x-3 )( 2x-3 ) $$

Step 1 :

After factoring out $ 2 $ we have:

$$ 4x^{2}-18x+18 = 2 ( 2x^{2}-9x+9 ) $$

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:

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

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 = -9 $.

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 $\color{blue}{ b = -9 }$

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

Step 7: Replace middle term $ -9 x $ with $ -3x-6x $:

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

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

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

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