Match the pattern of the scrap vinyl to the vinyl on the floor, and tape it down with duct tape. Make sure the tape holds the piece securely, but don't cover the pattern around the damaged area.
Using a straightedge and a sharp utility knife, cut along the lines of the pattern. Hold the utility knife straight up and down, and cut along the break of the line rather than the center of the pattern line. First, score the vinyl, and press firmly so the blade goes through the patch and the floor vinyl. It will probably take a few passes to cut through the vinyl.
Press the patch into place. Clean the seams with a damp rag, and lay a heavy object on the patch for the recommended drying time of the adhesive.
After the adhesive has dried, clean the seam of the patch, and mix a vinyl seam sealer. The sealer will melt the seam together so that it blends in with the rest of the floor.