Let [T] denote the dimension time and [L] distance (length).
The dimension equation is:
[L]=A[T]3+B[T]. Therefore A=[L][T]-3 and B=[L][T]-1 as dimensional quantities.
dx/dt=3At2+B, which dimensionally is:
[L][T]-1=[L][T]-3[T]2+[L][T]-1=[L][T]-1, a balanced equation dimensionally.