You said: "Those followers of extreme Islam beliefs can not be persuaded or negotiated with."
How do we know that? We haven't tried. We labeled them evil terrorists and decided to try and purge them from the earth. Your argument seems to have the basic assumption that huge percentages of the human population are clinically insane. Thats just not possible. We are fighting reasonable people from a very different culture with very different life situations. That makes them much harder to understand. It does not make them crazy. As humans they have the same wants needs and desires as you and I, it comes hardwired to just about everyone.
You cant go into a conflict with the fundamental belief that your enemy is crazy or evil. That shuts down all respect and dialog, which pretty much guarantees continued war, continued growing hatred, and ever increasing attacks on US soil. Instead you should consider that while you don't understand the sanity of their position, they most likely have one worth listening to.
I'm not pitching some Ghandi thing, but I think diplomacy has been entirely abandoned by labeling these people as terrorists and just deciding they shouldn't be allowed to live. The whole terrorist label means what anyway? That they kill civilians? We do that. That they use fear to manipulate a population? We do that too, remember "shock and awe"? Because they target the people who finance and sponsor the problems in their country? Yup, we do that too.
If you don't even try and understand the other side, you'll never be able to bridge the gap. This isn't a game of quake where you try to completely erase the other side, ultimately we're going to have to learn to get along (unless your pitching some kind of genocide?) or we'll end up like Israel with enemies on all sides and car bombings every day.
In terms of how to convert our enemies to our friends here are a few smart people have tried successfully in the past:
1) Show respect for their human rights
2) Acknowledge their legitimate grievances and concerns, and attempt to compromise with them where possible.
3) Show a willingness to help improve their quality of life (Try pouring 100 billion into housing developments, hospitals, and schools in Palestine and Iraq instead of bombing for example.)
4) Help them understand who we are and where we are coming from
5) Stop killing their families
6) Stop bulldozing their homes
7) Stop calling them evil
8) Stop treating them generally like they're primitives
If you or I as a sane person was in their situation I'm willing to bet we'd hate America. Depending on how directly we'd been impacted by the war we might even want to join the fight. After all, how many friends and family members of yours would have to die before you decided to fight back? How many of your homes would have to be bulldozed? How many of your dictators would you have to see propped up by the US? America has done much to earn the hatred of these people, its time we start addressing the root of the problem instead of spraying gas on the fire by continuing to kill people's relatives.
There absolutely are a handful of crazy people out there who mean us harm. This is nothing new, and they aren't specific to Muslim beliefs. (think Waco, Oklahoma City, Unabomber) Occasionally they'll get us, most of the time they'll be stopped. But you cant wage a war against a handful of crazy people from all walks of life, especially not by blunt force war that kills thousands and creates more enemies than it kills. I truly believe that just like everywhere else, people in the middle east would prefer to live in peace than in war... but you can only be stepped on so long before you have to act.
Since someone is probably thinking it by now... for the record I'm an agnostic white American male who's never been to the middle east.