\section*{Problem 2 - 271022}

Determine all natural numbers \(n\) for which

\[
\frac{n^{4}-2 n^{3}-3 n^{2}+7 n-6}{n+2}
\]

is a natural number!
}

Polynomial division of numerator and denominator yields

\[
\left(n^{4}-2 n^{3}-3 n^{2}+7 n-6\right):(n+2)=n^{3}-4 n^{2}+5 n-3
\]

The term \(a_{n}=n^{3}-4 n^{2}+5 n-3\) ( \(n\) natural number) grows strictly monotonically from a certain \(n_{0}\). It is given that \(a_{0}=-3, a_{1}=-1, a_{2}=-1, a_{3}=3, \ldots\) (2) and

\[
a_{n+1}-a_{n}=3 n^{2}-5 n+2>0 \quad \text { for } \quad n \geq 2 ; n_{0}=2
\]

Thus, (1) becomes a natural number for all natural numbers \(n \geq 3\).