= Solution
The prime <ideals> of $\mathbb R[x]$ are $(0)$ and the maximal <ideals> generated by the monic irreducibles
$$
x-a\quad(a\in\mathbb R),\qquad (x-a)^2+b^2\quad(a\in\mathbb R, b>0).
$$
For a nonzero prime $I=(p)$, use the residue classes of $1,x,\ldots,x^{N-1}$, where $N=n\deg p$. Multiplication by $x$ on $\mathbb R[x]/(p^n)$ is represented by the <companion matrix> of the monic <polynomial> $p^n$. In particular, for $p=x-a$ this may equivalently be written as one size-$n$ <Jordan block> with <eigenvalue> $a$. The <ideal> $(0)$ does not give a finite-dimensional quotient.
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