y=4sin^2x-sinx-1
=(2sinx)^2-sinx+(1/4)^2-(1/4)^2-1
=(2sinx-1/4)^2-17/16
∵x∈[-π/6,2π/3],
∴sinx∈[-1/2,1]
即2sinx∈[-1,2]
当2sinx=1/4时,y取最小值-17/16
当2sinx=2时,y取最大值2
y∈[-17/16,2]