\(\frac{3n+8}{n+1}=\frac{3\left(n+1\right)+5}{n+1}=\frac{3\left(n+1\right)}{n+1}+\frac{5}{n+1}=3+\frac{5}{n+1}\in Z\)
\(\Rightarrow5⋮n+1\)
\(\Rightarrow n+1\inƯ\left(5\right)=\left\{1;-1;5;-5\right\}\)
\(\Rightarrow n\in\left\{0;4\right\}\left(n\in N\right)\)