To say sorry, I'm terribly sorry and I'm really sorry are strong and warm. I apologize is a little more formal.
Apologizing
I'm terribly sorry — I'm terribly sorry for the wait.
I apologize — I apologize for the trouble.
I'm really sorry — I'm really sorry about this.
Try it: Amina: There is a problem with my room. Hiroshi: ___ What is wrong? Hiroshi works at the hotel and wants to apologize. Which fits? (I'm terribly sorry. / Excuse me, but / I'm not happy with) Answer: I'm terribly sorry. — I'm terribly sorry is an apology, which fits Hiroshi's reply. Excuse me, but and I'm not happy with are used to complain, so neither one apologizes.
Complaining politely
To complain in a polite way, start with Excuse me, but, then say There's a problem with or I'm not happy with before the thing that is wrong.
Excuse me, but — Excuse me, but this is cold.
There's a problem with — There's a problem with my room.
I'm not happy with — I'm not happy with the noise.
Try it: Sofia: How is everything in your room? Diego: ___ the noise. I can't sleep. Diego wants to complain politely about the noise. Which fits? (I'm not happy with / I apologize / I'm really sorry) Answer: I'm not happy with — I'm not happy with introduces a polite complaint, which fits. I apologize and I'm really sorry are both apologies, so neither one complains.