Find positive integers such that $2n^3 + 5|n^4 +n+1$ $$ 2n^3 + 5 | 2n^4 +2n +2 - 2n^4 - 5n$$
$$= 2n^3+5 | 2-3n$$
$$3n-2≥2n^3 + 5$$
is this correct? Is there a more efficient way?

Mod $(2n^3+5)$, you have that $2n^3\equiv-5$. Now suppose $(2n^3+5)$ divides $n^4+n+1$. Then mod $(2n^3+5)$:
$$
\begin{align}
n^4&\equiv-n-1\\
2n^4&\equiv-2n-2\\
n(2n^3)&\equiv-2n-2\\
-5n&\equiv-2n-2\\
-3n&\equiv-2\\
2-3n&\equiv0
\end{align}
$$
So $2n^3+5$ must divide $2-3n$. You can then test values of $n$ to arrive at the inequality.
