Answer
The quotient will be ${{x}^{2n}}-{{x}^{n}}+1$.
Work Step by Step
We have to find the quotient; let us divide the dividend by the divisor:
${{x}^{n}}+1\overset{{{x}^{2n}}-{{x}^{n}}+1}{\overline{\left){\begin{align}
& \,\,\,\,\,\,\,{{x}^{3n}}+1 \\
& \,\,\,\,\,\,\,{{x}^{3n}}+{{x}^{2n}} \\
& \frac{-\,\,\,\,\,\,\,\,-}{\begin{align}
& \,\,\,-{{x}^{2n}}+1 \\
& \frac{\begin{align}
& \,\,-{{x}^{2n}}-{{x}^{n}} \\
& \,\,+\,\,\,\,\,\,\,\,\,+ \\
\end{align}}{\begin{align}
& \,\,\,{{x}^{n}}+1 \\
& \,\,\,{{x}^{n}}+1 \\
& \frac{-\,\,\,\,\,-}{0} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}$
Therefore, the quotient will be ${{x}^{2n}}-{{x}^{n}}+1$.
