Mystery Starting Problem

I have a 04 Mitsubishi Galant that has occassionally been slow to start but eventually would over the last 3 months. Battery and alternator have been tested at a garage and numerous auto supply stores but both appear to be fine. Yesterday it actually didn’t start. I waited a little while and tried it again and it was again slow but, eventually started. I immediately took it to a garage. They’ve tried it repeatedly over the last day and half and couldn’t recreate the problem. They’ve tested again and see no issues with battery, alternator or starter. Any ideas?

The first idea is fuel flow reduced by a dirty fuel filter; but, on Mitsubishi (I think) the fuel filter is inside the fuel tank, and isn’t (normally) changed. The fuel flow, injected into the engine, could be too rich for starting.
When it doesn’t start, hold the gas pedal to the floor while cranking. As it starts (if it starts), ease up on the gas pedal. If this technique helps, it may be the engine coolant temperature sensor (ects) is causing it by sending the wrong temperature signal to the engine computer; or, the idle air control valve (iac) is the cause.
Or, the engine is too lean for start from low fuel flow, or fuel injector flow.
You, then, spray Starting Fluid into the large black intake tube. There’s, usually, a smaller hose you can disconnect from the intake tube and spray the Starting Fluid into. Attempt to start the engine. Tell your mechanic the results.

By “slow” do you mean the engine is turning over slowly?

Or, the engine turns over at a normal brisk RPM, but is slow to “fire up and run” on its own?

I don’t know the correct terminology to use so please bear with me. By slow, I mean, when I normallys start the car, it does so within seconds with no problem. During the times I’m describing, I turn the key in the ignitition and it sounds like it’s going to start but instead of doing so, I get a kind of whining sound and it eventually starts and in the one case it didn’t start at all.

Was it slow to start, today?

The whining sound you are hearing may be the starter motor turning but the bendix of the motor isn’t being pushing into the ring gear to turn the crank of the engine. Tell your shop you think the bendix of the starter motor needs to be worked on, or replace the starter if that is the case.

It stayed at the dealer overnight but they said they started it this morning and it was fine.

I just got the car back today because the Mitsubishi dealer service people still never found anything. I’ll try your suggestions now that I have it back.

Cougar,
I mistakenly overlooked this reply so, I didn’t tell the service people about the bendix and now I’ve picked up the car. If it happens again, I’ll tell them. Thanks so much for the advice.