Can I modify GPL software and include it in a proprietary software application?

You can modify GPL'd software and include it within a proprietary software program, you just can not distribute it. That is, a non-distributing end-user can do whatever they want with GPL'd software. The GPL does not require that all derivative works be under the GPL, only derivative GPL works that are distributed to others.

You can make private modifications to software licensed under the GPL and retain exclusively rights over your modifications as long as you do not distribute the software in source or binary form to a third party.