= Solution
Reduce coefficients modulo $I$:
$$
\Phi:R[X]\longrightarrow(R/I)[X],
\qquad
\sum_ka_kX^k\longmapsto\sum_k(a_k+I)X^k.
$$
This is a surjective <ring homomorphism> and its kernel consists exactly of <polynomials> all of whose coefficients lie in $I$, namely $I[X]$. The first isomorphism theorem gives the <coefficientwise quotient of a polynomial ring>
$$
\boxed{R[X]/I[X]\cong(R/I)[X]}.
$$
If $I$ is prime, then $R/I$ is an <integral> domain, so $(R/I)[X]$ is an <integral> domain. The quotient criterion therefore shows that $I[X]$ is prime in $R[X]$.
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