In my Analysis class, a section demonstrates how the extension field $\mathbb{R}[x]/(x^2+1)$ provides a root for $x^2 + 1$.
In an example leading up to this formal definition for the complex numbers, THIS EXAMPLE is given to familiarize the reader with the idea of an extension field providing a root to an irreducible polynomial $p(x)$.
While in the formal definition of imaginary numbers, $[x]$ is shown to be the $\sqrt{-1}$, in the attached example I can not make sense of $[x]$ as a root for $x^2 + x + 1$. In the formal definition of the complex numbers, the extension field $\mathbb{R}[x]/(x^2 + 1)$ represents the entire complex numbers. Does $K$ in the attached example represent just a portion of complex numbers? How can I determine $[x]$ in terms of complex numbers? What do I learn about $p(x)$ and its complex roots from what I have learned about its root in $K$?
I understand the theorems stating that field extensions contain a root of $p(x)$ but I do not understand how to determine what that root really is relative to $p(x)$.
Thank you!