Answer
$4x-7+\displaystyle \frac{4x+8}{x^{2}+x+1}$
Work Step by Step
$ \begin{array}{llllllllll}
& & 4x & -7 & & & & \color{red}{ \small{Quotient}} & & \\
& & -- & -- & -- & -- & -- & -- & & \\
x^{2}+x+1 & ) & 4x^{3} & -3x^{2} & +x & +1 & & & & \\
& & 4x^{3} & +4x^{2} & +4x & & & \color{red}{\leftarrow \small{ x^{2}(x^{2}+x+1) } } & & \\
& & -- & -- & -- & & & \color{red}{\leftarrow \small{ subtract } } & & \\
& & & -7x^{2} & -3x & +1 & & & & \\
& & & -7x^{2} & -7x & -7 & & \color{red}{\leftarrow \small{2x(x^{2}+x+1) } } & & \\
& & & -- & -- & -- & & \color{red}{\leftarrow \small{ subtract }} & & \\
& & & & 4x & +8 & & \color{red}{\leftarrow \small{ Remainder } } & &
\end{array} $
$\displaystyle \frac{4x^{3}-3x^{2}+x+1}{x^{2}+x+1}$= $4x-7+\displaystyle \frac{4x+8}{x^{2}+x+1}$
