Grammar
N5AI-drafted〜ませんか
won't you ...? / how about doing ...?
Formation
V-ます形（ますを取る）+ ませんか
Usage
This is the standard polite way to invite someone to do something together. Because it is phrased negatively, it sounds soft and leaves the listener free to refuse. Compare ～ましょうか, which assumes the other person already agrees or offers to do something for them; ～ませんか is the first invitation.
Examples
一緒にお茶を飲みませんか。
いっしょにおちゃをのみませんか。
Won't you have some tea with me?
土曜日、映画を見に行きませんか。
どようび、えいがをみにいきませんか。
How about going to see a movie on Saturday?
よかったら、うちで晩ご飯を食べませんか。
よかったら、うちでばんごはんをたべませんか。
If you like, why don't you have dinner at my place?