Which one is correct? And if both are, then please explain to me in which situations can I use "seems to have" and what is the grammar term for it? Thanks in advance.
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Both sentences are grammatically correct.
It seems like he asked her.
This is a comment on the situation as a whole ("it").
"What's going on? Can we go?"
-- "Well, it seems like he asked her, but I'm not sure what's happening."
He seems to have asked her.
This is a comment on the "he".
"What's going on? Can Harry come?"
-- "He seems to have asked her, but I'm not sure if he can come."
The primary difference between these sentences is that the first sentence is about it, whereas the second sentence is about he. Both are correct; it just depends on what the speaker is emphasizing.