ませんか
(Mase n ka)
won't you...? / would you like to...? (invitation).
Connection rules
- +Verb ます-stem + ませんか (the same stem used for the ます form).
- +Functionally an invitation, not a literal yes/no question despite the negative shape.
- +Compare with 〜ましょうか (shall we) — ましょうか assumes the other party is amenable, while ませんか leaves more room to decline.
Examples
1.
一緒に映画を見ませんか。

Issho ni eiga o mi mase n ka.
Would you like to watch a movie together?
「won't you...? / would you like to...? (invitation)」
2.
明日、一緒に食事しませんか。

Ashita, issho ni shokuji shi mase n ka.
Would you like to have a meal together tomorrow?
「won't you...? / would you like to...? (invitation)」
Practice
Would you like to drink some coffee together?
一緒にコーヒーを___。
Romaji also accepted