Factor polynomial $$ 2x^{2}+27x+81 $$

$$ 2x^{2}+27x+81 = ( 2x+9 )( x+9 ) $$

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:

$$ a = 2 , b = 27 ~ \text{ and } ~ c = 81 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 2 }$ by the constant term $\color{blue}{c = 81} $.

$$ a \cdot c = 162 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 162 $ and add to $ b = 27 $.

Step 4: All pairs of numbers with a product of $ 162 $ are:

PRODUCT = 162
1     162	-1     -162
2     81	-2     -81
3     54	-3     -54
6     27	-6     -27
9     18	-9     -18

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 27 }$

PRODUCT = 162 and SUM = 27
1     162	-1     -162
2     81	-2     -81
3     54	-3     -54
6     27	-6     -27
9     18	-9     -18

Step 6: Replace middle term $ 27 x $ with $ 18x+9x $:

$$ 2x^{2}+27x+81 = 2x^{2}+18x+9x+81 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 9 $ out of the last two terms.

$$ 2x^{2}+18x+9x+81 = 2x\left(x+9\right) + 9\left(x+9\right) = \left(2x+9\right) \left(x+9\right) $$

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