What are the maximal ideals of $\mathbb{R}[x]/(x^3)$?

Question

The way I see it, $\mathbb{R}[x]/(x^3)$ is ring whose elements are equivalence classes, which in their reduced form are of the form $ax^2+bx+c$ where $a,b,c\in\mathbb{R}$. I can see that $([x])$ ($[x]$ denotes the equivalence class of $x$ modulo $x^3$) and $([x^2])$ are ideals and my guess is that these are maximal. However, I cannot prove it (maybe we can find a field which is isomorphic to $(\mathbb{R}[x]/(x^3))/([x])$, but I cannot understand this nested quotient ring). Are there any other ideas I am missing?

Edit: grammatic correction

Answer 1

Note that $(x^2)$ is not prime since $x \notin (x^2)$ but $x \cdot x = x^2 \in (x^2)$. Therefore, $(x^2)$ is not maximal.

On the other hand, $(x)$ is a maximal ideal because $(\mathbb{R}[x] / (x^3)) / (x) \cong \mathbb{R}[x] / (x^3, x) = \mathbb{R}[x] / (x) \cong \mathbb{R}$ is a field.