PA-44 instructor pilot reported Right Hand Main Landing Gear Failed to Extend. The flight crew attempted to get the gear to lock in place; but were unable to do so. The flight crew elected to return to home airport and make a gear up landing.

Narrative

I flew with a student in a PA-44. We were practicing some takeoffs and landings at ZZZZ. We had done about 2-3 already and on the next one my Student put the Gear Down and I saw that the Right Main Gear Indicating Light was not Illuminated. We went around and I asked Tower if they could see the Gear they said it looked like the Right one was still up. We left their airspace to troubleshoot we did so by Cycling the Gear; using the Emergency Gear Extension; Resetting the Hydraulic Pump Circuit Breaker; pulling G's and the Right Main was still indicating up. We called the Owner of our Flight School who is also an A&P and he walked us through some procedures and when nothing was working he told us to head back to ZZZZ1 and he would call Tower and let them know our situation. Once at ZZZZ1 we started with a fly by and Tower told us our one of our Gear was still Up. We did a few circuits in the pattern before deciding to put the Gear Up and land with them All Up. We did so and afterwards spoke with the FAA and the NTSB did come out to inspect the plane as well.