Define S to be the following relation on R[x]: S = {(f, g) = R [x]²: (x + 1) divides (g - f) in R[x] }. You may assume that S is an equivalence relation. i. Prove that, if (f₁, f₂) = S and (9₁, 9₂) € S for some f₁, f2, 91, 92 € R [x], then also (f₁ + 9₁, f₂ +9₂) € S. ii. Suppose we wanted to prove that the set P of equivalence classes of S is a ring, similar to how we proved that Zm is a ring in the notes. Explain how part (i) would come up within the proof that P is a ring.

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