Answer
Quotient = $\quad x^{2}-\displaystyle \frac{2}{3}x-\frac{1}{9}$
Remainder = $\quad \displaystyle \frac{16}{9}x-\frac{17}{9}$
Work Step by Step
$\begin{array}{lllllll}
& x^{2} & -\frac{2}{3}x & -\frac{1}{9} & & & \\
& -- & -- & -- & -- & & \\
3x^{2}+x+1 \ \ ) & 3x^{4} & -x^{3} & +0 & +x & -2 & \\
& 3x^{4} & +x^{3} & +x^{2} & & & \\
& -- & -- & -- & -- & -- & \\
& & -2x^{3} & -x^{2} & +x & -2 & \\
& & -2x^{3} & -\frac{2}{3}x^{2} & -\frac{2}{3}x & & \\
& & -- & -- & -- & -- & \\
& & & -\frac{1}{3}x^{2} & +\frac{5}{3}x & -2 & \\
& & & -\frac{1}{3}x^{2} & -\frac{1}{9}x & -\frac{1}{9} & \\
& & & -- & -- & -- & \\
& & & & \frac{16}{9}x & -\frac{17}{9} & \\
& & & & & &
\end{array}$
Check:
$(3x^{2}+x+1)(x^{2}-\displaystyle \frac{2}{3}x-\frac{1}{9})+\frac{16}{9}x-\frac{17}{9}$
$=3x^{4}-2x^{3}-\displaystyle \frac{1}{3}x^{2} $
$\qquad \ \ \ \ +x^{3}-\frac{2}{3}x^{2}- \frac{1}{9}x\ $
$\ \ \qquad\qquad++x^{2}-\frac{2}{3}x-\frac{1}{9}+\frac{16}{9}x-\frac{17}{9}$
$=3x^{4}-x^{3}+x-2$
