grammar - "With what [...]?" or "What [...] with?"

Making a comparison with Who/Whom I now have a question about the use of what with prepositions in questions. I'll explain by example:

These two sentences are correct, one is more formal than the other:

Who/Whom

• Formal: With whom are you doing that?


• Informal: Who are you doing that with?

But if I substitute Who/Whom with What, the "formal" one sounds a bit awkward and I'm not sure if it's correct or not:

What

• With what are you doing that?


• What are you doing that with?

So, are those last two questions grammatically correct?
Is one more formal than the other?

Answer

Yes, they are both grammatical, but With what are you doing that? is much more formal, and probably quite rare.