What to ask or know before mortgage refinancing?

[Greg]

I got a call from the folks who did our mortgage, offering refinancing. The end result for us would be that our monthly payments would be lower, and we'd be resetting the 30-year clock. There are a whole 45 cents due at closing, though this needs to be run past the underwriters and a credit check perform before anything is locked in.

Is there anything I need to ask or know before we proceed with this?