Describe $\Bbb{R}[x]/(x^2 + 1)$ First by previous knowledge, I do know that $\Bbb{R}[x]/(x^2 + 1) \cong \Bbb{C}$, so this might seem trivial. But I am not here to ask about that and I don't want to use this fact
I am reading Dummit and Foote on Chapter 13 and this example (pg 515) describes the field as "an extension of degree $2$ of $\Bbb{R}$ in which $x^2 + 1$ has a root". I am not sure what this sentence means.
First there was a theorem that stated the basis elements of $K = \Bbb{R}[x]/(p(x))$ as described by $\theta \equiv  x\pmod{p(x)}$ and preceding paragraphs describe multiplication in the quotient ring.
Okay here is the actual question, they described the elements as $a + b\theta$ with $a,b\in \Bbb{R}[x]$. How do we arrive at that form? There is no talk about the isomorphism to $\Bbb{C}$ until much later. 
I do not not see how setting $x^2 + 1 \equiv 0$ turns $a_0 + a_1x + a_2x^2 + \dots a_nx^n\to a_0 + a_1x$
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In $\mathbb{R}[x]/(1+x^2)$, $[x]$ (that is, $x+(x^2+1)$) is a root of the equation $x^2+1=0$, since $[x]^2+1=[x^2+1]=0$.
And if $p(x)\in\mathbb{R}[x]$ there are polynomials $q(x),r(x)\in\mathbb{R}[x]$ such that $p(x)=q(x)\times(x^2+1)+r(x)$ and $\deg r(x)\leqslant 1$. So, $r(x)=a+bx$ for some real numbers $a$ and $b$ and$$\bigl[p(x)\bigr]=\bigl[r(x)\bigr]=[a+bx]=a+b[x].$$
