x*x-2009x+1=0的一个根是a
a^2-2009a+1=0
a^2+1=2009a
a-2009+1/a=0,a+1/a=2009
(a*a-2008a)+2009/(a*a+1)
=(a^2-2009a+1)+a-1+2009/2009a
=0+a-1+1/a
=2009-1
=2008