ZAB Controller; providing OJT; experienced an airspace incursion event when inaccurate vectors and confused coordination resulted in an aircraft crossing airspace boundaries.

Narrative

Aircraft X was cleared off 89TE 'as filed' (89TE..SJT..JEN...) climbing to 170 by me. I was giving OJT at the time to a brand new trainee; but I was asked by my supervisor to provide an experienced R-Side trainee OJT. I gave a relief briefing and the trainee assumed the sector. Shortly after I began R-Side OJT; Aircraft X called our frequency as having departed 89TE. After some time getting him to RADAR contact (he departed on a VFR code); my trainee climbed the aircraft to FL270 and subsequently issued a 15 degree left turn to assist the aircraft in staying clear of Mexican airspace. I was fairly complacent at the situation because the sector was slow; and the R-Side trainee has multiple sectors around the specialty; so I wasn't paying close attention to the situation. H98 (RSG LO sector) called and informed us that Aircraft X was RADAR contact. My trainee said that he had already shipped the aircraft; and the Houston Controller stated that they had him on frequency but did not have the hand off. Compounding the situation; the vector given to the aircraft I had assumed would keep the plane well clear of Mexican Airspace; but in fact he did cross the border; deviating MMTY Airspace. During discussions with the trainee; he stated that he flashed the aircraft specifically to H98; but he thought it auto-flashed to H97. URET and the data block indicated that H97 had taken the hand off. It was his belief that ZHU had turned the aircraft toward their airspace; ergo violating us as we had never released control. After revisiting the situation; however; it seems apparent that we climbed the aircraft to FL270; flashed him to H97 (which owns FL280 and above) and failed to 1.) Monitor the RADAR vector to ensure separation from MMTY; and 2.) Point out the aircraft to either MMTY or H98. Basically; I didn't pay close enough attention my trainee to see the progression of the incident; and I believe we subsequently violated MMTY and (potentially) H98 Airspace. Recommendation; trainers pay more attention to even 'experienced' R-Side trainees. Watch closely to see what aircraft are actually doing; particularly in the realm of hand off status.