= Solution
Here $J_4=(X^2+1)$ is a <principal ideal>. The <evaluation homomorphism> $p\mapsto p(i)$ from $\mathbb R[X]$ onto $\mathbb C$ has this <kernel of a ring homomorphism>: division by $X^2+1$ leaves a remainder $a+bX$, and $a+bi=0$ with real $a,b$ forces both to vanish. Surjectivity follows because every complex number is $a+bi$. Consequently
$$
\boxed{\mathbb R[X]/J_4\cong\mathbb C,\qquad J_4\text{ is maximal}.}
$$
Equivalently, $X^2+1$ is an <irreducible polynomial> over the <real numbers>.
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