Since Exp[a] Exp[b] automatically evaluates to
Power[E, Plus[a, b]],
it's impossible to keep Exp[a] Exp[b] as is, unless you use Hold[] or HoldForm[].
On the other hand, we have this:
Factor[Exp[a + b] + c Exp[a + d]]
(* E^a (E^b + c E^d) *)
If a, b, c, d have numeric values, then you have to take a more manual approach, since the numbers will combine with one another. Try the following:
Exp[a] Expand[(Exp[a + b] + c Exp[a + d])/Exp[a]]
