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