Consider the ring R = Z[i], the ring of Gaussian integers, and the polynomial ring R[x]. a) Prove that R[x] is a Euclidean domain. Provide a detailed Euclidean function and demonstrate the Euclidean algorithm within R[x]. b) Determine whether R[x] is a Principal Ideal Domain (PID) and justify your answer. c) Investigate the irreducibility of the polynomial f(x) = x² + 1 in R[x]. Provide a comprehensive proof of its reducibility or irreducibility. d) Assuming f(x) is reducible, factorize it into irreducible polynomials in R[x]. If it is irreducible, explain the implications for the ring R[x]/(f(x)).