Answer
Quotient $=x^2-2x+\dfrac{1}{2}$
Remainder $=\dfrac{5x+1}{2}$
Work Step by Step
The given expression is
$(2x^4-3x^3+x+1)\div(2x^2+x+1)$
Rewrite the expression as
$(2x^4-3x^3+0x^2+x+1)\div(2x^2+x+1)$
Perform long division:
$\begin{matrix}
& x^2 & -2x & +1/2 & & && \leftarrow &\text{Quotient}\\
&-- &-- &--&--& \\
2x^2+x+1) &2x^4&-3x^3&+0x^2&+x&+1 & \\
& 2x^4 &+x^3 & +x^2& &&& \leftarrow &x^2(2x^2+x+1) \\
& -- & -- & --& &&&\leftarrow &\text{subtract} \\
& 0 & -4x^3& -x^2 &+x & & \\
& &-4x^3 & -2x^2&-2x &&& \leftarrow &-2x(2x^2+x+1) \\
& & -- & --& -- &&&\leftarrow &\text{subtract} \\
& & 0 & +x^2& +3x &+1 & & \\
& &&x^2 & +x/2&+1/2 && \leftarrow &1/2(2x^2+x+1) \\
& && -- & --& -- &&\leftarrow &\text{subtract} \\
& && 0 & +5x/2& +1/2 & & \leftarrow & \text{Remainder}
\end{matrix}$
Checking:
$\text{(Quotient)(divisor)+ Remainder}$
$=(x^2-2x+1/2)(2x^2+x+1)+5x/2+1/2$
$=2x^4-4x^3+x^2+x^3-2x^2+x/2+x^2-2x+1/2+5x/2+1/2$
$=2x^4-3x^3+x+1$
$=\text{ Dvidend}$
Hence, the quotient is $x^2-2x+\frac{1}{2}$ and the remainder is $\dfrac{5x+1}{2}$.
