You're a waiter at a restaurant. A customer is complaining about her meal, but she's already eaten it so you can't take it back. You want her to be satisfied, so you offer a free dessert:
May I offer you a complimentary dessert?
This is a very formal and polite way to offer something to someone. Waiters, flight attendants, and other people in service jobs speak this way:
May I offer you today's newspaper?
May I offer you a dessert menu?
You can also use this phrase if you have guests in your home or office who you don't know very well:
May I offer you something to drink?
"Complimentary" means "free", but it sounds a lot fancier and nicer. So hotels, restaurants, theaters, and stores use this word when they want to offer something free to their customers without sounding too cheap. Here are examples of "complimentary" things:
two complimentary tickets
a complimentary breakfast
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