x^2-3x+1=0
x^2+1=3x
(2x^5-5x^4+2x^3-8x^2)/(x^2+1)
=[(2x^5-6x^4+2x^3)+(x^4-3x^3+x^2)+(3x^3-9x^2+3x)-3x]/3x
=[(2x3+x^2+3x)(x^2-3x+1)-3x]/3x
=-3x/3x
=-1