解f(x)=sin^2x+2sinxcosx+3cos^2x,
=sin^2x+cos^2x+2sinxcosx+2cos^2x
=1+sin2x+2cos^2x
=2+sin2x+2cos^2x-1
=sin2x+cos2x+2
=√2sin（2x+π/4)+2
函数的对称轴满足2x+π/4=kπ+π/2,k属于Z.
故函数的对称轴为x=kπ/2+π/8,k属于Z.
函数图像的对称中心满足2x+π/4=kπ,k属于Z
故函数图像的对称中心满足x=kπ/2-π/8,k属于Z,
故函数的对称中心为（kπ/2-π/8,0）,k属于A.