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.