Let $R:=\mathbb R[x]/\langle (x^2+1)^2 \rangle$ and $J:=\langle x^2+1 \rangle \lhd R$. Prove that every finitely generated $R$-module is isomorphic to $R^m \oplus (R/J)^n$ for a unique pair of non negative integers $(m,n)$.
My first doubt is if $\langle x^2+1 \rangle \subset R$ or of $\mathbb R[x]$? I suppose they meant $\langle \overline{x^2+1} \rangle$. Since $R$ is a PID and we are under the hypothesis that $M$ is finitely generated, then $$M \cong R^m \oplus R/\langle {p_1}^{\alpha_1} \rangle \oplus \cdots\oplus \langle {p_n}^{\alpha_n} \rangle$$ where $p_i$ is prime and $\alpha_i \in \mathbb N$ (by the way, are the $m$ and $\alpha_i$ unique?).
I am lost with the exercise, any suggestions would be appreciated.
$R$ is not a PID, instead it is a PIR (since $\mathbb R[x]$ is a PID) and moreover artinian and local. Now you can apply the result from this topic. Note that the only ideals of $R$ are $(0)$, $J$, and $R$.