I get four solutions with Solve this way:
sols = Solve[LHS == RHS, \[Alpha]1]
The solutions are to be verified. Here is how they behave with a choice of the parameters:
paramValues = {c1 -> 1, c2 -> 1, M1 -> 2, M2 -> 3};
sols /. paramValues
% // N
LHS == RHS /. sols /. paramValues
