It believe it is correct, but whether it is acceptable to an instructor depends on your course. You may have studied an algorithm to prove similar claims, and then you need to go over the steps without justifying them. However, for this to look like a proof to an outsider, it should include not only computational steps, but also invocation of some lemmas that prove that the computations indeed show what you claim they do.
Euclid's lemma for polynomials. Therefore, it is sufficient to check whether the denominator is divisible by these two polynomials. Then one should say a word or two why does not divide and does not divide .