Outline your pattern with blue painter's tape. You can do a simple border around the edges, or get creative and make an interesting geometric design. You may want to work with a stencil or template to get more complex designs just right. Even if you decide to add a design, a border along the edges will make your carpet remnant look finished and more like a proper area rug. Since paint will change the texture of the carpet, it's better to create smaller designs than to paint the majority of the remnant. Let the paint be an accent rather than the main event.
Apply the paint to your carpet using a paint roller. The thinner your design, the smaller your paint roller should be. Roll the paint on evenly without applying it too thickly. Latex paint works well on harder carpets like Berber or sisal. If you're using a softer carpet, like wool or cotton, opt for acrylic paints specially formulated for fabrics. They won't cause your carpet to dry as stiffly as some other paints will.
Allow the paint to dry completely. Latex paint will take at least two hours to dry, but you may want to leave it overnight just to be on the safe side. Once the paint is dry, remove the tape. Peeling away the tape will reveal your stunning new design.