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Your ring S is not an Euclidean domain, hence sage does not know what you mean by gcd. R2 is an Euclidean domain. You can lift elements of S to elements of R2 using the lift() method, e.g.:

sage: xgcd(f.lift(), g.lift())
(1, b^5 + 2*b^3 + 2*b^2 + 2, 2*b^5)