Factor polynomial $$ 2x^{2}+9x+9 $$

$$ 2x^{2}+9x+9 = ( 2x+3 )( x+3 ) $$

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:

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

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

$$ a \cdot c = 18 $$

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 $ 6x+3x $:

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

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

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

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