[verb て-form] + いただけませんか [verb て-form] + いただけないでしょうか お + [verb ます-stem] + いただけませんか ご + [noun] + いただけませんか
ていただけませんか is a polite request, "could you please ~?". It goes after the て-form of a verb, and it is what you say to a teacher, a customer, a stranger or someone senior at work.
いただく is the humble verb for "receive", and いただける is its potential form. Word for word you are asking "can I not receive your doing ~?", so the request is about what you get, not what the other person does.Among friends it sounds stiff; use てくれないきごう? there.
The potential いただける is what turns the sentence into a request. Without it the sentence no longer asks anything.
These go beyond the て-form request.
In shops and business email the て-form is often replaced by お + the ます-stem, or ご + a する-noun.
Without the negative the request is still polite, but a little more direct.