Which of the following antibiotics would be appropriate for a patient with gram-positive organisms and who is resistant to penicillin?

Nafcillin is the correct choice for a patient with gram-positive organisms that are resistant to penicillin. Nafcillin is a penicillinase-resistant penicillin, specifically designed to overcome bacterial resistance mechanisms associated with staphylococcal infections. This antibiotic is effective against certain staphylococci, including those that produce penicillinase, an enzyme that can inactivate many penicillins, including standard penicillin.

The other options may not be suitable for use with resistant gram-positive organisms. Cephalexin is a first-generation cephalosporin that can work against some gram-positive bacteria but may not be effective against strains resistant to penicillin, particularly those producing penicillinase. Carbenicillin, while effective against some gram-negative bacteria, is less effective against resistant gram-positive organisms and is not typically the drug of choice for such infections. Amoxicillin, a derivative of penicillin, would also be ineffective against organisms that have developed resistance to penicillin, as it shares similar susceptibility limitations.

Therefore, nafcillin's resistance to the inactivation by penicillinase makes it the most appropriate choice in managing infections caused by resistant gram-positive organisms.