WOW. There is a lot packed into this question.

First: an improper fraction is one where the numerator (top of the fraction) is larger than the denominator (bottom of the fraction).

Next: if the fraction is NOT improper (proper) then you can still use long division.

Next: in your example NOTICE that x+1 is ALWAYS greater than x! (add anything to x and it will be greater than x), so it is always proper.

Next: in order to get an answer (quotient) to the division problem where there is NO REMAINDER), the quotient Must be a factor of the dividend (whats inside the division box). therefore..... you are right that the top must be bigger than the bottom!

I think you need to review rational expressions or get tutoring in rational expressions. (and Factoring).

Example: look at 3/x. sometimes it will be improper and sometimes it won't. Ya gotta know what x is.

The "KEY" to you question ( if I understand it correctly) lies in factoring (finding roots).

I tutor online and in home.