Suppose $f, g \in Q[x]$ non-zero elements. Let $(f, g) = (d),$ and $h \in (d).$ Then there exit polynomials $p,q$ such that $h =pf + gf.$ I want to show that $h= p^{\prime}f +q^{\prime}g$ iff $p^{\prime} = p + s \frac{g}{d}$ and $q^{\prime} = q - s \frac{f}{d}$ for some $s \in Q[x].$
I think I am almost done. So far I have the following equation: $$ (p-p^{\prime})f = -(q-q^{\prime})g.$$ From here I want to conclude $((p-p^{\prime}) \in (\frac{g}{d}).$ I am stuck. Could you help me? Thanks so much.
By the way, only part is trivial. I just need to prove the if part.
You are almost there. Since $d$ is a gcd of $f$ and $g$, $f/d$ and $g/d$ are coprime. Now divide your equality by $d$ ans apply Gauss lemma to conclude.