OK. My idea is as follows:

Theorem (Munkres p161). A space X is locally path connected if and only if for every open set U of X, each path component of U is open in X.

B is locally path-connected by the above theorem. Let path components of B be . Then , where each is simply-connected. We see that is not necessarily simply-connected. However, if we apply the smash product of A and B, is identified for . Now each simply-connected space has a common point. It follows that the smash product is simply-connected. We see that a suspension of a simply-connected space is simply connected.

Thus we concluce that the suspension of a smash product is simply connected having a trivial fundamental group.