Don't confuse soundproofing with room treatment. Soundproofing is to kill the sound you make outside the room. Room treatments are to take care of problem frequencies in your mixing environment to provide a flat response with no reflections.
Room treatments are done by strategically placing materials in specific places in the room to absorb unwanted frequencies and standing waves.
Sooundproofing is an entirely different thing, and is very difficult and expensive to achieve. Basically, there are only two methods- using mass, and decoupling. Basically, you would use the mass of heavy sheetrock, a gap and then another wall of heavy sheetrock. Along with this, you need to look for ways to decouple the walls so the sound vibrations don't pass through floor joists, ceilings, etc.
Soundproofing and room treatments are something you'd really need to study up on. Room treatments don't have to be expensive, but soundproofing almist always is.
Egg crates and carpet won't do it, bro.