In　this　technical　note,　the　problem　of　designing　fixed-order　robust　H-infinity　controllers　is　considered　for　linear　systems　affected　by,　polytopic　uncertainty.　A　polynomial　method　is　employed　to　design　a　fixed-order　controller　that　guarantees　that　all　the　closed-loop　poles　reside　within　a　given　region　of　the　complex　plane.　In　order　to　utilize　the　freedom　of　the　controller　design,　an　H-infinity　performance　specification　is　also　enforced　by　using　the　equivalence　between　robust　stability　and　H-infinity　norm　constraint.　The　design　problem　is　formulated　as　a　linear　matrix　inequality　(LMI)　constraint　whose　decision　variables　are　controller　parameters.　An　illustrative　example　demonstrates　the　feasibility　of　the　proposed　design　methods.