ませんか

ませんか is a polite, inviting way to suggest or invite someone to do something. It softens the request and expects agreement rather than giving a command. It commonly follows a verb in the ます-form to turn it into a tentative question. It can be used to propose activities, seek confirmation, or invite someone to join you. Examples: いっしょに映画を見ませんか？ (Would you like to watch a movie together?) 食事に行きませんか。 (Shall we go out to eat?) 今日は忙しいですか。 is not the typical usage; instead, use 今日は忙しくないですか to ask if someone is busy. Note that this form is appropriate in informal to polite conversations, and slightly more formal than plain ませんか in casual settings.