There is one extra square bracket at the end. If it's deleted, then:
Show[
LineIntegralConvolutionPlot[{{y, -y - 2 x - 2}, {"noise", 500, 500}}, {x, -3, -1}, {y, -4, 4}, ColorFunction -> "Rainbow",
LightingAngle -> 0, LineIntegralConvolutionScale -> 3, Frame -> False],
LineIntegralConvolutionPlot[{{y, -y}, {"noise", 500, 500}}, {x, -1, 1}, {y, -4, 4}, ColorFunction -> "Rainbow", LightingAngle -> 0, LineIntegralConvolutionScale -> 3, Frame -> False],
LineIntegralConvolutionPlot[{{y, -y - 2 x + 2}, {"noise", 500, 500}}, {x, 1, 3}, {y, -4, 4}, ColorFunction -> "Rainbow",
LightingAngle -> 0, LineIntegralConvolutionScale -> 3, Frame -> False],
PlotRange -> {{-3, 3}, {-4, 4}}]