I don't think you can enter (5/x - 7/(x + 1)) ((x + 1) - x) and keep ((x + 1) - x) from evaluating to 1. So you'll have to enter or evaluate some other expression.

Of course, if it's allowed to enter something else, then entering -12 + (7 x)/(1 + x) + (5 (1 + x))/x will evaluate to the desired expression. This suggestion probably seems flip, but if something different is to be entered, I am far from understanding what differences are allowed and what differences are prohibited. Any help?

I also do not understand what sequence of algebraic operations you have mind that lead from (5/x - 7/(x + 1)) ((x + 1) - x) to -12 + (7 x)/(1 + x) + (5 (1 + x))/x. I could probably reverse-engineer something if I thought long and hard. Frankly, though, I don't think it's worth my time, when you probably already have something in mind that you could share.

This might be a good start: Inactivate[(5/x - 7/(x + 1)) ((x + 1) - x)]. Then you might be able to perform the structural operations you have mind. When done, then apply Activate[] to the transformed expression.

Gee, I accidentally I discovered it:

foo = Inactivate[(5/x - 7/(x + 1)) ((x + 1) - x)]
Distribute[foo, Inactive@Plus, Inactive@Times] // Activate

The operation involves treating ( x + 1 ) (x+1) and x x as separate entities and multiplying each with the terms inside the other parentheses, rather than computing ( x + 1 − x ) (x+1−x) first—which is how your code implements it. Thank you very much!