Finite Field with GF(2^8)

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I am curious how I would go about solving this problem. I would show some work but I have no idea where I would start with this problem. If someone could give me any direction it'd be greatly appreciated, or even just a source I can read that will point me in the right direction. I've been trying to do some research on my own but many of the sources seem to be different than what is happening here.

Thank you so much