The subsets of {b, d} are {}, {b}, {d}, {b, d}. Thus the subsets of interest to you are
... {a, e}, {a, b, e}, {a, d, e}, {a, b, d, e}
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We know your final sets will include {a, e}, so they're not part of the variable portion of the subsets. We know c is not included, so we can leave it out of consideration. That leaves only b and d included (or not) in the variable portion of the sets of interest. We only needed to find all the ways that we could have b or d or neither or both, that is, the subsets of {b, d}. Then we form the union of each of those subsets with {a, e}.