a) Show that if there exists $n \in \mathbb{N}$ such that $a n^{2}+b n+c=0$, then $n \mid c$

b) Determine the integers $n \in \mathbb{N}$ such that $n^{5}-2 n^{4}-7 n^{2}-7 n+3=0$.
a) For $a n^{2}+b n+c=0 \Longleftrightarrow c=-n(a n+b)$, hence it is required that $n \mid c$
b) We apply the previous result: we have $n \mid 3$ so $n \in\{-3,-1,1,3\}$. Conversely, only -1 and 3 are suitable.