I know that it's hard to solve symbolically AREs or in general quadratic matrix equations, some considerations on the symbolic solution of matrix equation AX^2+BX+C=0 can be found in Higham & Kim, "Numerical analysis of a quadratic matrix equation" IMA j. numer. anal. 20 (2000), no. 4, 499--519. You can use computer algebra systems to solve polynomial equations.
In alternative you could consider numerical solution, there are quick solvers for particular Riccati equation and they need only matrix multiplication and inversion. It depends on the dimension of matrix. Please send your equation in the form XDX+A'X+XA-C=0.
Bye Bruno Iannazzo