Solution
= Solution
The evaluation map $\varphi:\mathbb R[x]\to\mathbb C$, $p\mapsto p(i)$, is a <ring homomorphism> and is onto because $a+bx$ maps to any $a+bi$. By <polynomial division>, write
$$
p(x)=(x^2+1)q(x)+a+bx.
$$
Then $p(i)=0$ precisely when $a=b=0$. Thus $\ker\varphi=(x^2+1)$, and the <first isomorphism theorem> gives
$$
\boxed{\mathbb R[x]/(x^2+1)\cong\mathbb C.}
$$
Since the <quotient ring> is a <field>, \b[$(x^2+1)$ is a maximal ideal].