Putting a good question out there, see who's got good answers...

What do you do when you're out with a girl and she becomes very engaged in a conversation with another guy who she either slightly knows or has just approached her? Is there a polite way to handle this without amoging?

As in a gf, a girl you just approached, or a date?

Lets say a date

Had this happen to me before act completely unphased by it walk up nicely and politely, and most importantly confidently introduce yourself shake his hand say hey babe I'm gonna order a drink do you want anything. I've been in that situation 3 times all with dates, and its worked perfectly all 3 times. The reasons I think it worked so well is 1) the guys she's talking to may become intimidated because you are not threatened by him at all. 2) the girl finds your confidence attractive. I've seen where guys feel threatened they approach trying to look alpha, but you can see right through their scharade.

Anyway that's my opinion and what has wroked for me good luck!!

You can also try to get yourself involved and listen to the guy. Then give him all the validation he wants. Say things like "Wow! Really!?" and "No way!?". Tone it down after a while and become more and more quiet. Finally, turn to the girl and ask her something like "do you want to go and have a drink?", and end the conversation with the guy.

If a girl talks on the phone or with someone while we're out, I just hit on and talk to other girls.
Warning, because "relationships" have ended this way. I still think it's the best thing to do.