a是方程x^2-3x+1=0的根,
則a^2-3a+1=0；
a^2-2a=a-1；a^2+1=3a.
則2a^2-5a-2+2/(a^2+1)
=2（a^2-2a）-（a+2）+2/(3a)
=2（a-1）-（a+2）+2/(3a)
=a-3+2/(3a)
=[3（a^2+1)-1]/(3a) -3
=(3*3a-1)/(3a) -3
=-1/(3a).
解方程x^2-3x+1=0得:x=a=(3±√5)/2;
則2a^2-5a-2+2/(a^2+1)=-1/(3a)=(-2/3)/(3±√5)
=(3±√5)/6.