Answer
Quotient = $\quad x^{2}-2x+0.5$
Remainder = $\quad 2.5x+0.5$
Work Step by Step
$\begin{array}{lllllll}
& x^{2} & -2x & +0.5 & & & \\
& -- & -- & -- & -- & & \\
2x^{2}+x+1 \ \ ) & 2x^{4} & -3x^{3} & +0 & +x & +1 & \\
& 2x^{4} & +x^{3} & +x^{2} & & & \\
& -- & -- & -- & -- & -- & \\
& & -4x^{3} & -x^{2} & +x & +1 & \\
& & -4x^{3} & -2x^{2} & -2x & & \\
& & -- & -- & -- & -- & \\
& & & x^{2} & +3x & +1 & \\
& & & x^{2} & +0.5x & +0.5 & \\
& & & -- & -- & -- & \\
& & & & 2.5x & +0.5 & \\
& & & & & & \\
\end{array}$
Check:
$(2x^{2}+x+1)(x^{2}-2x+\displaystyle \frac{1}{2})+2.5x+0.5$
$=2x^{4}-4x^{3}+x^{2}+x^{3}-2x^{2}+0.5x+x^{2}-2x+0.5+2.5x+0.5$
$=2x^{4}-3x^{3}+x+1$
