ませんか details
write + won't you…?
kaki masen ka
won't you…?; how about…?
meaning: won't you write
帰りにラーメンでも食べていきませんか？
kaeri ni raamen demo tabete ikimasen ka?
Want to grab some ramen on the way home? neutral
今度の日曜、一緒に美術館へ行きませんか。
kondo no nichiyou, issho ni bijutsukan e iki masen ka.
This Sunday, won't you go to the museum with me? neutral
Examples
won't you…? — inviting the listener to join or do something
帰りにラーメンでも食べていきませんか？
kaeri ni raamen demo tabete ikimasen ka?
Want to grab some ramen on the way home? neutral
今度の日曜、一緒に美術館へ行きませんか。
kondo no nichiyou, issho ni bijutsukan e ikimasenka.
This Sunday, won't you go to the museum with me? neutral
よろしければ、来週の見学会に参加しませんか。
yoroshikereba, raishuu no kengakukai ni sankashimasenka.
If you are interested, won't you join next week's tour? formal
Formation
{V-masu-stem}ませんかattaches to ます-stem
Notes
Related patterns
Similar
手伝いましょうか offers the speaker's help, 手伝いませんか asks for the listener's — ましょうか proposes what the speaker will do or what is already on the table, ませんか recruits the listener into something new.
ましょう announces joint action as already settled and invites the listener aboard; ませんか genuinely asks and leaves a clean way to decline — which is why invitations aimed upward prefer it.
Lookalikes
The same 〜ませんか negative-question frame at speed — but 行きませんか invites the listener into joint action, while てくれませんか's てくれ routes the benefit to the speaker, making 手伝ってくれませんか a favor request, not an invitation.