Factor polynomial $$ 5x^{2}-54x+81 $$

$$ 5x^{2}-54x+81 = ( x-9 )( 5x-9 ) $$

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:

$$ a = 5 , b = -54 ~ \text{ and } ~ c = 81 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 81} $.

$$ a \cdot c = 405 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 405 $ and add to $ b = -54 $.

Step 4: All pairs of numbers with a product of $ 405 $ are:

PRODUCT = 405
1     405	-1     -405
3     135	-3     -135
5     81	-5     -81
9     45	-9     -45
15     27	-15     -27

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -54 }$

PRODUCT = 405 and SUM = -54
1     405	-1     -405
3     135	-3     -135
5     81	-5     -81
9     45	-9     -45
15     27	-15     -27

Step 6: Replace middle term $ -54 x $ with $ -9x-45x $:

$$ 5x^{2}-54x+81 = 5x^{2}-9x-45x+81 $$

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

$$ 5x^{2}-9x-45x+81 = x\left(5x-9\right) -9\left(5x-9\right) = \left(x-9\right) \left(5x-9\right) $$

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