I think your best bet would be one of the shuttle services; either Prime Time Shuttle or SuperShuttle. The fare from the Carnival Terminal in Long Beach to LAX is $32 total for two in a shared van; but the shuttle will take you to any hotel in the area. The distance to LAX is 22.79 miles and the estimated time is 30 minutes.
The Carnival transfer would be much more expensive and more inconvenient too. Taxis are plentiful but also very expensive.
Another advantage of the two shuttle services I mentioned: you can reserve your ride and pay in advance: one less hassle upon disembarkation.