= Solution
Use the reduction criterion: a primitive integer-coefficient <polynomial> whose leading coefficient remains nonzero modulo a <prime number> $p$ is irreducible over $\mathbb Q$ if its reduction is irreducible over $\mathbb F_p$. This follows from <Gauss lemma for polynomials>, since a nontrivial factorization over the integers would reduce to a nontrivial factorization of the same degrees. For a monic polynomial the factors may be taken monic, so neither degree can drop.
Modulo $2$ the polynomial is $\overline f=t^4+t+1$. Its values at $0$ and $1$ are both $1$, so it has no <linear polynomial> factor. A reducible <quartic polynomial> without a <linear polynomial> factor must factor into two irreducible <quadratic polynomials>. The only monic irreducible <quadratic polynomial> over $\mathbb F_2$ is $q=t^2+t+1$: among monic <quadratic polynomials> a nonzero constant is necessary, and $t^2+1$ has a root, while $q$ has none.
But $t^2\equiv t+1\pmod q$, so $t^3\equiv1$ and $t^4\equiv t$. Consequently
$$
\overline f\equiv t+t+1=1\pmod q.
$$
Thus $q$ does not divide $\overline f$, excluding the remaining possible factorization. We obtain
$$
\boxed{t^4+t+1\text{ is irreducible over }\mathbb Q.}
$$
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