= Solution
The polynomial $p=x^5-4x+2$ satisfies the <Eisenstein criterion> at the prime two: every nonleading coefficient is even, the leading coefficient is not, and the constant term is not divisible by four. Hence it is irreducible over $\mathbb Q$. Its ideal is maximal in $\mathbb Q[x]$; explicitly, for any polynomial not divisible by $p$, the Euclidean algorithm gives $ag+bp=1$, furnishing the inverse of its residue class. Thus
$$
\boxed{\mathbb Q[x]/(p)\text{ is a field}.}
$$
For the integer quotient, division by the monic polynomial $p$ gives a unique remainder of degree below five with integer coefficients. If an integer polynomial becomes zero in the rational quotient, its integer remainder is still divisible by $p$ over $\mathbb Q$, so must be zero. This gives an injective ring map $\mathbb Z[x]/(p)\to\mathbb Q[x]/(p)$, proving the <monic irreducible integer-polynomial quotient is a domain>.
The same unique-remainder argument makes the integer quotient a free abelian group with basis $1,x,x^2,x^3,x^4$. The class of two is nonzero and cannot have an inverse: twice an integer remainder cannot equal the remainder one. Consequently
$$
\boxed{\mathbb Z[x]/(p)\text{ is an integral domain but not a field}.}
$$
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