I think "rip the rug" may refer to the common idiom 'rip the rug from under somebody/something'. Rip the rug from under someone means: to suddenly take away help or support from someone, or to suddenly do something which causes many problems for them So in this sentence I assume it means that driving away would somehow damage her reputation. I hope this helps :)

This sounds like a rough transcript of David Cameron's speech about the recent Scottish referendum. His quote is, "If we lost Scotland, if the UK changed, we would rip the rug from our own reputation." The part "if we drive her away" sounds like a paraphrase. Either that, or I'm barking up the wrong tree. Anyway... Think of a person standing on a rug. Someone grabs the rug and pulls it sharply - rip! Imagine what happens to the person who was standing on the rug a moment ago. To be honest, it's an odd choice of phrase. I wonder if Mr Cameron understood it properly.