Answer
$x^2-2x+\frac{1}{2}+\frac{\frac{5}{2}x+\frac{1}{2}}{2x^2+x+1}$
Work Step by Step
We use the long division as shown in the figure below.
Checking:
$$
(2x^2+x+1)(x^2-2x+\frac{1}{2} )+\frac{5}{2}x+\frac{1}{2}\\
= 2x^4-4x^3+x^2+x^3-2x^2+\frac{x}{2}+x^2-2x+\frac{1}{2}+\frac{5x}{2}+\frac{1}{2} \\
=2x^4-3x^3+x+1
.$$
