 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