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?

Using the Extended Euclidean Algorithm (which can be applied to $\mathbb{Q}[x]$ as well as $\mathbb{Z}$), we get
$$
\left(36n^2+24n+16\right)\left(n^4+n+1\right)-\left(18n^3+12n^2+8n-27\right)\left(2n^3+5\right)=151
$$
if $2n^3+5\mid n^4+n+1$, we must also have $2n^3+5\mid151$. Since $151$ is prime and there is no integer $n$ so that $2n^3+5=\pm1$ or $2n^3+5=\pm151$, there is no integer $n$ so that $2n^3+5\mid n^4+n+1$.
