So what is the real reason for the Gaza Invasion?

My guess is that they just didn't see any alternative. The Israelis can't possibly want a battle in a densely populated urban setting with a very hostile population and a motivated and well-armed force of irregulars defending from familiar positions. Despite that, the Israelis have been remarkably successful in killing a high ratio of Hamas fighters, about 46% of the dead so far. I doubt that was the motivation though. Hopefully both sides will enter into a lasting ceasefire promptly that results in no more missile attacks on Israel and a relaxed border for Gaza. Much easier said than done I am afraid.