Finding congruence solutions in F[x] and checking my work

[Joe20]

Hi all, I have completed this question as attached. Hope someone could help to check if my solutions are correct. However, I am not sure what theorem is used for part (d). May need some advice. Thanks.

 

[Euge]

All your solutions look correct. (Yes)

Just a minor comment, but you might want to put the equivalence class brackets around the terms belonging to $R$.