(Would you) care to join us?
Do you want to join us? Tom and Mary saw Fred and Sally sitting at another table in the restaurant. Tom went over to them and said, "Would you care to join us?" Mary: Isn't that Bill and Sue over there? John: Yes, it is. Shall I ask them to join us? Mary: Why not? John (after reaching the other table): Hi, you guys! Care to join us? Bill: Love to, but Sue's mom is going to be along any minute. Thanks anyway.
(Would you) care to...?
a polite phrase introducing an inquiry as to whether someone wishes to do something. John: Would you care to step out for some air? Jane: Oh, I'd love it. Sue: Care to go for a swim? Mary: Not now, thanks.