Actually.. hate to disagree but egg cartons do virtually nothing to prevent sound from leaving a room.
They do a minimal job of reducing room reflections and they do nothing on low frequencies. You might be able to cut back a bit of mic feedback or deaden the room slightly for recording, but that's about it.
There's only two ways to stop sound from escaping a room and that's with distance and mass. You need something like a second wall that doesn't touch the outside wall and insulation between them.