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\space\space\space$ ${x^2+x-1}$
$x^2-x+1$$\space)\overline{x^4+0x^3-x^2+0x+1}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{x^4\space-x^3\space+x^2}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space x^3-2x^2+0x$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{x^3\space-x^2\space+x}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-x^2\space-x+1$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\underline{-x^2\space+x-1}$
$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space-2x+2$
Check answer:
$(x^2+x-1)\cdot(x^2-x+1)-2x+2$
$x^4-x^3+x^2+x^3-x^2+x-x^2+x-1-2x+2$
$x^4-x^2+1\checkmark$
