= Solution
<Eisenstein's criterion> states that an integer-coefficient <polynomial> is irreducible over $\mathbb Q$ if some <prime number> divides every nonleading coefficient, does not divide the leading coefficient, and its square does not divide the constant coefficient. Equivalently apply this to the primitive part when phrased for primitive polynomials.
Take $p=2$. Every nonleading coefficient of $t^5+4t+2$ is divisible by $2$ (including the zero coefficients), its <leading coefficient> is $1$, and its constant coefficient $2$ is not divisible by $4$. Therefore
$$
\boxed{t^5+4t+2\text{ is irreducible over }\mathbb Q.}
$$
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