If $L_1L_2$ is regular language then $L_2L_1$ is regular to?
I was posting only a hint, then I saw other full answers, so this is a full (hidden) succinct solution :-)


   Let $L_1 = \{ 1^p \mid p \text{ is prime}\}$, $L_2 = \{ 1^* 0 \}$; we have $L_1 L_2 = \{ 11^+ 0\}$ which is regular, but $L_2 L_1 = \{ 1^* 0 1^p \mid p \text { is prime}\}$ which is not regular.