Find the quotient when $x^4 + 3x^2 - 7x$ is divided by $x^2 + 2x - 1.$
The long division is shown below.

\[
\begin{array}{c|cc ccc}
\multicolumn{2}{r}{x^2} & -2x & +8 \\
\cline{2-6}
x^2 + 2x - 1 & x^4 & & +3x^2 & -7x &  \\
\multicolumn{2}{r}{x^4} & +2x^3 & -x^2 \\
\cline{2-4}
\multicolumn{2}{r}{} & -2x^3 & +4x^2 & -7x \\
\multicolumn{2}{r}{} & -2x^3 & -4x^2 & +2x \\
\cline{3-5}
\multicolumn{2}{r}{} & & +8x^2 & -9x & \\
\multicolumn{2}{r}{} &  & +8x^2 & +16x & -8 \\
\cline{4-6}
\multicolumn{2}{r}{} &  &  & -25x & +8 \\
\end{array}
\]Thus, the quotient is $\boxed{x^2 - 2x + 8}.$