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Fill in the Blank #3?

I have never been good with these kind of puzzles, and I am completely lost on this one. It feels a little too much like guess what's on the author's mind. Anyways, can someone give me a hint on how it works?

I struggled on it too, couldn't figure it out. I had to ask the author for the solution.
Here's a clue: Pair the top-bottom, pair the left-right, and find the relationship of those two-digit numbers with the middle digit.

That's quite a hint. :) I wonder how many figured that one out...

I can solve the first one as a polynomial sequence, but is that the correct solution, and is there a nicer explanation? For the second one, the same approach works, but there I found a very simple rule.