Factor polynomial $$ 50x^{2}+45x+9 $$

$$ 50x^{2}+45x+9 = ( 10x+3 )( 5x+3 ) $$

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:

$$ a = 50 , b = 45 ~ \text{ and } ~ c = 9 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 50 }$ by the constant term $\color{blue}{c = 9} $.

$$ a \cdot c = 450 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 450 $ and add to $ b = 45 $.

Step 4: All pairs of numbers with a product of $ 450 $ are:

PRODUCT = 450
1     450	-1     -450
2     225	-2     -225
3     150	-3     -150
5     90	-5     -90
6     75	-6     -75
9     50	-9     -50
10     45	-10     -45
15     30	-15     -30
18     25	-18     -25

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 45 }$

PRODUCT = 450 and SUM = 45
1     450	-1     -450
2     225	-2     -225
3     150	-3     -150
5     90	-5     -90
6     75	-6     -75
9     50	-9     -50
10     45	-10     -45
15     30	-15     -30
18     25	-18     -25

Step 6: Replace middle term $ 45 x $ with $ 30x+15x $:

$$ 50x^{2}+45x+9 = 50x^{2}+30x+15x+9 $$

Step 7: Apply factoring by grouping. Factor $ 10x $ out of the first two terms and $ 3 $ out of the last two terms.

$$ 50x^{2}+30x+15x+9 = 10x\left(5x+3\right) + 3\left(5x+3\right) = \left(10x+3\right) \left(5x+3\right) $$

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