\(2n^3-n^2+7n-1⋮n^2+3\)
\(\Leftrightarrow2n\left(n^2+3\right)-\left(n^2+3\right)+n+2⋮n^2+3\)
\(\Leftrightarrow\left(n^2+3\right)\left(2n-1\right)+n+2⋮n^2+3\)
Vì \(\left(n^2+3\right)\left(2n-1\right)⋮n^2+3\)
\(\Rightarrow n+2⋮n^2+3\)
\(\Leftrightarrow\left(n+2\right)\left(n-2\right)⋮n^2+3\)
\(\Leftrightarrow n^2-4⋮n^2+3\)
\(\Leftrightarrow n^2+3-7⋮n^2+3\)
Vì \(n^2+3⋮n^2+3\Rightarrow7⋮n^2+3\)
\(\Rightarrow n^2+3\inƯ\left(7\right)=\left\{7\right\}\)( vì \(n^2+3\ge3\forall x\))
\(\Leftrightarrow n^2=4\)
\(\Leftrightarrow n=\left\{\pm2\right\}\)
Thay vào thấy \(x=-2\)thỏa mãn
Vậy \(x=-2\)
\(2n^3-n^2+7n-1⋮n^2+3\Leftrightarrow2n^3-n^2-2n\left(n^2+3\right)+7n-1⋮n^2+3\left(vìx\in Z\right)\Leftrightarrow2n^3+7n-2n^3-6n-n^2-1⋮n^2+3\Leftrightarrow n-n^2-1⋮n^2+3\Leftrightarrow n-n^2-1+n^2+3⋮n^2+3\Leftrightarrow n+2⋮n^2+3\left(1\right)\Leftrightarrow n\left(n+2\right)⋮n^2+3\left(vìn\in Z\right)\Leftrightarrow n^2+2n⋮n^2+3\Leftrightarrow n^2+2n-n^2-3⋮n^2+3\Leftrightarrow2n-3⋮n^2+3\left(2\right)\) \(\left(1\right);\left(2\right)\Rightarrow\left\{{}\begin{matrix}n+2⋮n^2+3\\2n-3⋮n^2+3\end{matrix}\right.\Leftrightarrow2n+4-\left(2n-3\right)⋮n^2+3\Leftrightarrow7⋮n^2+3.Mà:n^2+3\ge0+3=3\Rightarrow n^2+3=7\left(vì:n\in Z\right)\Rightarrow n^2=4\Leftrightarrow n=\pm2\)
