jmc

The long division is shown below.

$$\begin{array}{cccccc}\text{multicolumn}2r{x}^2& +3x& +4& & & \\ \text{cline}2-6x^2-3x+5& x^4& & & & +1\\ \text{multicolumn}2r{x}^4& -3x^3& +5x^2& & & \\ \text{cline}2-4\text{multicolumn}2r& +3x^3& -5x^2& & & \\ \text{multicolumn}2r& +3x^3& -9x^2& +15x& & \\ \text{cline}3-5\text{multicolumn}2r& & +4x^2& -15x& +1& \\ \text{multicolumn}2r& & +4x^2& -12x& +20& \\ \text{cline}4-6\text{multicolumn}2r& & & -3x& -19& \end{array}$$Thus, the remainder is $\boxed{-3x-19}.$