How Do I Lengthen a Video Clip to Match the Same Duration As a Piece of Music?

You can use a Time Warp effect on the video in After Effects and reduce the speed setting to lengthen the clip. Other video editing or compositing programs have similar Time effects that can be applied, but may be under a different name. Keep in mind that video is only a series of images that are displayed at a high rate to create the illusion of movement, and unless you shot the footage with a high speed camera, the more you slow the speed of the clip, the closer it will come to looking like a slide show.