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RegionPlot[ And @@ Thread[(Re[sol /. z -> (a + b I)]) < 0], {a, 3, 10}, {b, -10, 10}]

Mathematica graphics

Edit

As per your edit, z is Real (really bad choice for a name), so:

Plot[Re[sol], {z, -1, 10}, Evaluated -> True, PlotRange -> {-2, 2}]

Mathematica graphics

You can see that from z==4 onwards, the two complex conjugate solutions are negative, as well as the real one.

RegionPlot[ And @@ Thread[(Re[sol /. z -> (a + b I)]) < 0], {a, 3, 10}, {b, -10, 10}]

Mathematica graphics

RegionPlot[ And @@ Thread[(Re[sol /. z -> (a + b I)]) < 0], {a, 3, 10}, {b, -10, 10}]

Mathematica graphics

Edit

As per your edit, z is Real (really bad choice for a name), so:

Plot[Re[sol], {z, -1, 10}, Evaluated -> True, PlotRange -> {-2, 2}]

Mathematica graphics

You can see that from z==4 onwards, the two complex conjugate solutions are negative, as well as the real one.

1
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RegionPlot[ And @@ Thread[(Re[sol /. z -> (a + b I)]) < 0], {a, 3, 10}, {b, -10, 10}]

Mathematica graphics