By the rational root theorem the only possible rational roots are $\pm 1, \pm 2$, and by inspection none of these are roots. If the polynomial is reducible, it therefore factors into the product of a quadratic and cubic factor (over $\mathbb{Z}$ by Gauss's lemma).
$\bmod 2$ the polynomial factors as $x(x^4 + x + 1)$. The latter factor has no root $\bmod 2$, so if it is reducible it is the product of two irreducible quadratics. But the only irreducible quadratic $\bmod 2$ is $x^2 + x + 1$, and $(x^2 + x + 1)^2 = x^4 + x^2 + 1$. Hence $x^4 + x + 1$ is irreducible $\bmod 2$.
But if the polynomial factored as the product of a qudaratic and cubic factor over $\mathbb{Z}$, it would only have at most cubic irreducible factors $\bmod 2$; contradiction. Hence the polynomial is irreducible.