Step 1 :

After factoring out $ 5 $ we have:

$$ 45x^{2}-165x+5075 = 5 ( 9x^{2}-33x+1015 ) $$

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:

$$ a = 9 , b = -33 ~ \text{ and } ~ c = 1015 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 9 }$ by the constant term $\color{blue}{c = 1015} $.

$$ a \cdot c = 9135 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 9135 $ and add to $ b = -33 $.

Step 5: All pairs of numbers with a product of $ 9135 $ are:

PRODUCT = 9135
1     9135	-1     -9135
3     3045	-3     -3045
5     1827	-5     -1827
7     1305	-7     -1305
9     1015	-9     -1015
15     609	-15     -609
21     435	-21     -435
29     315	-29     -315
35     261	-35     -261
45     203	-45     -203
63     145	-63     -145
87     105	-87     -105

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -33 }$

Step 7: Because none of these pairs will give us a sum of $ \color{blue}{ -33 }$, we conclude the polynomial cannot be factored.

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