If A and B are symmetric matrices then (AB+ BA) is :

given A = A^T

B = B^T

so  (AB + BA)^T = (AB)^T + (BA)^T

(as (A + B)^T = A^T + B^T)) 

⇒  B^TA^T + A^TB^T

(as  (PQ)^T = Q^TP^T)

⇒  BA + AB

so (AB + BA)^T = (AB + BA)

symmetric matrix