Answer
$\displaystyle \frac{3x^{4}+5x^{3}+7x^{2}+3x-2}{x^{2}+x+2}=3x^{2} +2x-1$
Work Step by Step
$ \begin{array}{llllllllll}
& & 3x^{2} & +2x & -1 & & & \color{red}{ \small{Quotient}} & & \\
& & -- & -- & -- & -- & -- & -- & & \\
x^{2}+x+2 & ) & 3x^{4} & +5x^{3} & +7x^{2} & +3x & -2 & & & \\
& & 3x^{4} & +3x^{3} & +6x^{2} & & & \color{red}{\leftarrow \small{ 3x^{2}(x^{2}+x+2) } } & & \\
& & -- & -- & -- & & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & 2x^{3} & +x^{2} & +3x & -2 & & & \\
& & & 2x^{3} & +2x^{2} & +2x & & \color{red}{\leftarrow \small{2x(x^{2}+x+2) } } & & \\
& & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract }} & & \\
& & & & -x^{2} & -x & -2 & & & \\
& & & & -x^{2} & -x & -2 & \color{red}{\leftarrow \small{ -1(x^{2}+x+2) } } & & \\
& & & & -- & -- & -- & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & & & & 0 & \color{red}{\leftarrow \small{ Remainder } } & &
\end{array} $
$\displaystyle \frac{3x^{4}+5x^{3}+7x^{2}+3x-2}{x^{2}+x+2}=3x^{2} +2x-1$
