“Could I sleep on your couch for a few nights?”
You are visiting Miami for business, but you don't want to stay in a hotel. You have a good friend from university who lives there, who has a big, comfortable couch.
Could I sleep on your couch for a few nights?
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a few (of something)
"A few" is a number that's not specific, but it usually means somewhere between 3-10. It's a little less formal than "several" and also sounds like a slightly lower number. Here are some examples:
You may want to consider hiring a personal trainer for a few sessions.
That's a very generous offer. Um, let me think it over for a few days and get back to you.
Could I (do something)?
This is a polite way to ask for something. It’s better than “Can I (do something)?” if you’re worried about being rude.
Could I get some water, please?
Could I come with you?
A "couch" is a soft piece of furniture that two or more people can sit on together.
For most English speakers, the words "couch" and "sofa" mean the same thing. You can use either word. For people who are really interested in furniture and interior design, there are some differences between "couch" and "sofa", which you can read about by searching for "couch vs. sofa" on the Internet.