Is $P(x)=x^n+2x^{n-1}+3x^{n-2}+\cdots+nx+(n+1)$ irreducible in $\mathbb{Z}[x]$ for all $n$?
Approach.
Lemma. From the book: Edmund Landau. Darstellung und Begrundung einiger neuerer Ergebnisse der Funktionentheorie.
Applying the lemma we have that all the roots have a module $|\alpha|>1$. If $n+1$ is prime then by contradiction (Vieta's formulas) $$P(x)=(x^r+\cdots\pm 1)(x^s\cdots\pm p)$$ it is shown that $P(x)$ is irreducible, but if $n+1$ is not prime $P(x)$ is irreducible?
Any hint would be appreciated.
