Answer
quotient $x^4-x^2+x+1$ and remainder $r(x)=\frac{3}{x-2}$
Work Step by Step
Step 1. The coefficients of the dividend can be identified in order as $\{1,-2,-1,3,-1,1\}$ and the divisor is $x-2$; use synthetic division as shown in the figure to get the quotient and the remainder.
Step 2. We can identify the result as $\frac{x^5-2x^4-x^3+3x^2-x+1}{x-2}=x^4-x^2+x+1+\frac{3}{x-2}$ with the quotient as $x^4-x^2+x+1$ and the remainder as $r(x)=\frac{3}{x-2}$
