So I have two examples from class that are causing me some confusion. We’re using the following fact (stated as Corollary 3.20(b) in our notes): $[a]_{I}=[b]_{I}$ iff $a-b\in I$ iff $a=b+x$ for some $x\in I.$
Example 1:
Example 2: Let $R=\mathbb{R}[X]$ and $I=\langle X^2+1\rangle$. Then for $a,b,c,d\in \mathbb{R}$ we have $$[a+bX]_{I}=[C+dX]_{I}$$ iff $a+bX = c+dX + (X^2+1)g(X)$ for some $g(X)\in R$. So $a=c$ and $b=d$ and thus $$R/I=\{[a+bX]_{I}:a,b\in \mathbb{R}\}$$ gives all the elements of $R/I$ without repeats.
So what I don’t understand is why neither of these two examples follows a similar format or the fact (Corollary 3.20(b)) stated above. For the first example I would expect $[a+bi]_{I}=[c+di]_{I}$ iff $(a+bi)-(c+di)\in I$ iff $a+bi = (c+di) + x$ for some $x \in I$ (with “$a=a+bi$” and “$b=c+di$”). For the second I'd have thought it'd be $a+bX=c+dX+g(X)$ for some $g(X)\in I.$
Where am I going wrong?