Answer
Quotient: $x^2-x-1$
Remainder: $2x+2$
Work Step by Step
$\space$$\space$$\space$$\space$$\space$$\space$$\space$$\space$$\space$ $\space$$\space$$\space$$\space$$\space$$\space$$\space$$\space$ $x^2-x-1$
$x^2+x+1$$\space)\overline{x^4+0x^3-x^2+0x+1}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{x^4\space+x^3\space+x^2}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-x^3-2x^2+0x$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{-x^3-x^2\space\space-x}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-x^2+x+1$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{-x^2-x-1}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space2x+2$
Check answer:
$(x^2-x-1)\cdot(x^2+x+1)+2x+2$
$x^4+x^3+x^2-x^3-x^2-x-x^2-x-1+2x+2$
$x^4-x^2+1\checkmark$
