jmc

The long division is shown below.

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