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She will not object to his delivering the speech as long as the speech makes sense.

I think him sounds better but my teacher said it was his, but she didn't explain why. Do you know why we should use his here and not him?

thank you!
Comments 
'Him' is increasingly common in informal speech, but it is still wrong when writing carefully. 'She' is not concerned about him; she is concerned aout his speech delivery. Compare:

I don't like him.
I don't like his dog.
I don't like his talking during class.
What if it's female?

Would it be

I don't like her.

vs

I don't like her walking to the park.

vs

I don't like her talking during nap time.
Students: Are you brave enough to let our tutors analyse your pronunciation?
Well, the 2 sentences are not 'vs' (opposed to each other). They both use the possessive pronoun (her/his walking, her/his talking). But yes, the female pronoun has the same form (her/her) while the others have different forms (me/my, him/his, them/their, us/our, you/your), and that may be one of the factors leading all of the object pronouns to become more common in this situation.