Is there a way of getting the airport extreme to see the Airport Express Base Station such that I can extend the range of my Wi-Fi using the Base Station?
You have it backwards. The AirPort Express needs to "see" the wireless signal that the AirPort Extreme is producing to be able to extend the network.
But first.....a few questions to see if things might work.
Does the AirPort Express have only one Ethernet port? If yes, look on the side of the device to find the model number. It's hard to see......light gray lettering on a white background. The model number will either be A1084, A1088, or A1264. Post back with the model number that you see.
Does the AirPort Express have two Ethernet ports? If yes, reply that you have the two Ethernet port version.
Will the AirPort Express connect to the main AirPort Extreme using a permanent wired Ethernet cable connection?
Or, will the AirPort Express connect to the AirPort Extreme using a wireless connection?
There are some significant downsides to this......the speed of the Wi-Fi network will be cut in half and the Express must be located where it can pick up a strong Wi-Fi signal from the AirPort Extreme.
Normally, this means that the Express is located about halfway between the AirPort Extreme and the area where you need more wireless coverage. But, you have to watch out for heavy obstructions like ceilings or bathroom tile that the signal will try to pass through. It will likely take some experimentation to find the right location for the Express.
Finally, remember that most items that are given away are done so for a reason. They are not working well, or not working at all.
