x²+3x+1=0,
x^2+1=-3x,
x+1/x=-3,
x^2+2+1/x^2=9,
x^2+1/x^2=7,
x+1/x=3,
x^2+1/x^2=7,
x^2/(x^4+x^2+1)
=1/(x^2+1+1/x^2)
=1/8.