Let R be UFD, Why can't we directly compute gcd of 2 polynomials in the R[x]?

Question

I encountered the following theorem in my algebra lecture notes, which gives a way to compute the gcd of 2 polynomials from R[x] using the quotient field of R and R itself, I don't understand why running the Euclid's algorithm in the R[x] itself wouldn't give me the gcd: The theorem from notes