• Would you please bring your library books back today as they are needed by another borrower?
• Could you join us on Saturday? Tom's back from Sydney and we're having a barbecue.
• Would you like to join us on Saturday? We're having a barbecue in the back garden.
• Would you care to join us on Saturday? We're celebrating Tom's return from Sydney.
• Do you want to join us on Saturday? We're having a bash in the garden.

Would you like…? + infinitive / Would you mind…?If you want to sound particularly polite, or if you think the answer may be negative, you can also use Would you mind + verb-ing as the preferred alternative to Could you…? Would you mind…? literally means: Would you object to…?

• Would you mind locking the door when you leave? ~ No, not at all!
• Could you please lock the door when you leave? ~ Yes, certainly!
• If you're not busy at the moment, would you mind helping me with my homework?
• If you're not busy at the moment, could you give me a hand with my homework?

Can I / could I / may I / might I If you are requesting something for yourself, all of these forms are possible. May and might are considered to be more polite, more formal or more tentative than can and could, but can and could are usually preferred in normal usage. Compare the following:

• Can I ask a favour of you? ~ Of course you can.
• Could I ask you to collect Deborah from school tomorrow ~ Of course you can.
• Could I possibly have another cup of coffee? ~ I don't think you should. You won't sleep tonight if you do.
• If you've finished with the computer, may I turn it off? ~ Yes, please do.
• Might I leave work a bit earlier today? I've got a doctor's appointment at 5.

• I wonder if I might leave work a bit earlier today? I've got a doctor's appointment.
• Would I be able to leave work a bit earlier today? I've got a dentist appointment at 6.
• Would it be OK if I left work a bit earlier today? I've got to take our cat to the vet.

Would you / Do you mind if I…? Similarly, if we use Do / Would you mind if I…? to make a request, we may be anticipating possible objections:

• Would you mind if I put off talking to Henry until tomorrow? ~ I think that's a mistake. I think you should speak to him today

• Would you mind filling the ice trays and putting them in the freezer? (= you do it)
• Would you mind me/my filling the ice trays and putting them in the freezer? (= I'll do it)