Prove that two different concatenations of relations are equivalent
This is best carried out using notations from algebra of relations.

You're asked to prove $RSRRSR \leq RSRSRS$. That $R$ is transitive means $RR\leq R$, so  $RSRRSR \leq RSRSR$ by monotonicity of composition. That $S$ is reflexive means $I\leq S$, where $I$, the identity relation, is neutral element for composition: $I\leq S$ entails $(RSRSR)I\leq (RSRSR)S$ (monotonicity of composition again). Finally, you have

$(RSR)^2 = RSRRSR \leq RSRSR = RSRSRI \leq RSRSRS = (RS)^3$

as required.