y=(3x²+3x+3+1)/(x²+x+1)
=3(x²+x+1)/(x²+x+1)+1/(x²+x+1)
=3+1/(x²+x+1)
x²+x+1
=(x+1/2)²+3/4>=3/4
所以0<1/(x²+x+1)<=4/3
3<3+1/(x²+x+1)<=3+4/3
所以最大值=13/3