N5
ませんか
Won't you, Would you not, Why don't we
ラーメンを食べに行きませんか。
Won't you go eat ramen with me?
Explanation
「〜ませんか」 is a polite invitation: 「一緒に映画を見に行きませんか」 (Would you like to go to a movie together?). The form is a negative question ('won't you …?'), but it means an invitation. Asking it negatively leaves the other person room to refuse, so it does not feel pushy, and it is more polite and indirect than 「〜ましょう」. It often goes with 「一緒に」.
Mixed up with
To accept, do not say 「いいえ」; say 「ええ、行きましょう」 or 「いいですね、ぜひ」. Saying 「いいえ、行きません」 is a refusal.
To refuse, Japanese people usually do not say 「行きません」; they trail off and give a reason, leaving the sentence unfinished: 「すみません、明日はちょっと…」, and the other person understands.