Can the polynomial in the numerator of the expression x^2-5x+7/x-9 be factored to derive (x-9) as a factor?

The answer is no.

Explanation #1
Define the numerator as
f(x) = x^2 - 5x + 7.

If (x-9) is a factor, then x=9 will be a zero of f(x), so that f(9)=0.
f(9) = 9^2 - 5*9 + 7 = 43
Because f(9)≠0, (x-9) is not a factor of f(x).

Explanation #2
Perform long division.

                     x   +  4
     
x-9 | x^2  -  5x  +  7
        x^2  -  9x
       
                   4x  +  7
                   4x  - 36
                 
                           43

Because there is a remainder of 43, the numerator cannot be factored with (x-9) as a factor.

What is the inequality?