네 노트북을 쓸 수 있어? vs 노트북 좀 써도 됩니까?
Which one is correct in the context below? (I know the politeness level of 노트북 좀 써도 됩니까? is different - just asking about the grammar in general)

A: 네 노트북을 쓸 수 있어?
B: 아니, 지금 내가 쓰고 있어. 조지한테 물어봐.
A: 조지야, 지금 노트북 쓸 거야? 내 노트북은 집에 있어. 네 것을 쓸 수 있어?
C: 응, 써. 선생님한테 이메일 보낼 거야?
A: 아니, 내 어머니한테 보낼 거야.

A: Can I use your laptop?
B: No, I am using it right now. Ask George.
A: George, are you going to use your laptop now? My laptop is at home. Can I use yours?
C: Yes, you can. Are you going to send an email to your teacher?
A: No, I will send it to my mother.

Question

네 노트북을 쓸 수 있어? vs 노트북 좀 써도 됩니까?
Which one is correct in the context below? (I know the politeness level of 노트북 좀 써도 됩니까? is different - just asking about the grammar in general)

A: 네 노트북을 쓸 수 있어?
B: 아니, 지금 내가 쓰고 있어. 조지한테 물어봐.
A: 조지야, 지금 노트북 쓸 거야? 내 노트북은 집에 있어. 네 것을 쓸 수 있어?
C: 응, 써. 선생님한테 이메일 보낼 거야?
A: 아니, 내 어머니한테 보낼 거야.

A: Can I use your laptop?
B: No, I am using it right now. Ask George.
A: George, are you going to use your laptop now? My laptop is at home. Can I use yours?
C: Yes, you can. Are you going to send an email to your teacher?
A: No, I will send it to my mother.

Answer

네 노트북을 쓸 수 있어?:
This sentence is vague, and can be understood in many different ways depending on the context:
Is your laptop available?
Is your laptop operational?

노트북 좀 써도 됩니까?:
This one is the more explicit and polite way asking for a "permission".
"Do you mind / Is it fine by you if I use your laptop for a while?"

- Clearly the better pick

Answer

I would say "노트북 좀 써도 괜찮아?" or "노트북 좀 써도 돼?"  to friends, and "노트북 좀 써도 됩니까?" to older people or in business situation as it sounds way more polite.