First off, if your the maintenance man, you should already know how to tackle this but, I'll give the benefit of the doubt here. Secondaly, Electrical Contractor is incorrect.
To provide you an accurate answer, I would have to know what type of fixture your dealing with but, will cover the repair for each.
If you have a T12 fixture, then there will be one ballast that runs the two outside bulbs and a one ballast that runs the two inside bulbs. If you've changed out both bulbs then were 1/3 the way there. Chances are that one of those two ballasts are bad. First inspect the tombstones (the part where the bulb twists into at each end). The typical problem here is that these wires stab into the tombstone and sometimes come loose but your problem is very typical of a bad ballast. To determine which ballast is bad, you'll have to turn on the light and cut the black wire to one of the two ballast and see if the inside lights go out. If the inside lights stay on then you've identified the bad ballast, otherwise it's the opposite ballast that you want to replace. Just make sure you use insulated wire cutters and don't be touching any metal on the cutters. Also make sure the feed side of the black wire doesn't stray to the housing or yourself and short out the circuit. It will require 8 blue wire nuts to change out the ballast and cut your wires close to the ballast so you'll have some slack to rewire the new ballast. Again, the probable cause is a bad ballast on a T12 fixture.
If your fixutre is a T8 or T5, then there is only one ballast and these can go bad rendering and odd or even number of bulbs not working, unlike the T12 ballast. In this case you'll have to replace the single ballast, again after inspecting the tombstones as these too are stab in connections.
Not all ballast are the same. There are 120V & 277V ballast, there are electronic ballast and magnetic ballast.
The voltage of the ballast are most often multi-voltage i.e.-120/277V. However, you'll have to make sure you purchase the like ballast when it comes to magnetic versus electronic or you will have to rewire the fixture per the wiring diagram on the new ballast. So to reorder the ballast, you'll need to know the voltage (most likely 277V), the bulb size T12, T8 or T5, the type of ballast (magnetic or electronic), and number of bulbs the ballast runs (2 lamp or 4 lamp).
Tempature of the ballast can also let you know which is bad and which is good. As I recall a the cooler of the two ballast will be the bad one (on a T12 fixture). The ballasts are located under the center cover and can be remove by sqeezing it on both sides at each end.
Tombstones are a pain to inspect so I usually replace the ballast first and then go for the tombstones last. I usually have a case of ballasts and really don't worry about wasting a ballast and just make sure you leave enough lead wire on the old ballast to reuse in the event that it is a bad tombstone. Broken plastic housing on a tombstone does not necessarily require replacing as long as it will hold the bulb. Due to the wide variety of tombstones, you may have to take that with you when you purchase new ones. The 3 typical tombstones types will be enclosed double pin, open double pin and single pin, each in a T12, T8 & T5 size. On a 4ft. fixture your most likely dealing with the open double pin. For the other two pin styles, they are spring loaded and you push them to one end and drop down the other end to remove the bulb. These applications are commonly reserved for the 8ft. bulb fixtures.