x^2-2x+2=(x-1)^2+1; x^2+2x+1=(x+1)^2, so 2x+1=(x+1)^2-x^2; x^2-(x-1)^2=2x-1. (But none of these help to do the division!)
(2x+1)/(x^2-2x+2) is a fraction because the highest power of x in the denominator is higher than that in the numerator, and the quadratic has no real factors. The division would lead to negative powers of x in the quotient and the quotient would continue indefinitely with increasingly negative powers of x.
The question would make more sense if the dividend and divisor were interchanged; x^2-2x+2 divided by 2x+1 to give a quotient and remainder using algebraic long division:
2x+1 | x^2 -2x+2 | x/2-5/4 rem 13/4. The quotient x/2-5/4 can also be expressed (2x-5)/4.
...........x^2+x/2
.................-5x/2+2
.................-5x/2-5/4
..........................13/4