Final answer:
The correct phrasal verb to use in the context of arriving and registering at a hotel is 'checked into'. The sentence should read: 'We had some problems when we checked into the hotel.'
Step-by-step explanation:
The correct phrasal verb to fit into the sentence 'we had some problems when we checked ___the hotel. they had reserved the room under the wrong name' is 'into'. The sentence should read: 'We had some problems when we checked into the hotel. They had reserved the room under the wrong name.' The term 'checked in' is commonly used to indicate that you have arrived at a hotel and are registering your details, typically by providing your information and obtaining your room key.
Learn more about Phrasal Verbs