could you would you mind

There are many different ways of making polite requests in English.
Would you…? / could you…? / would you like to…?

If you are asking other people to do things, you would normally use would you or could you + infinitive. would you like to…? is also a very polite way of suggesting or requesting something, politer than Do you want to…? Compare the following:

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?