Final answer:
Pseudomonas aeruginosa is the gram-negative organism among the options provided that can cause gram-negative infective endocarditis, which is less common than infections caused by gram-positive bacteria.
Step-by-step explanation:
Among the organisms listed, Pseudomonas aeruginosa is a gram-negative bacterium that can cause gram-negative infective endocarditis. Although it is less common than gram-positive organisms like Streptococcus viridans, Staphylococcus aureus, and Enterococcus faecalis, which are typical causative agents of endocarditis, Pseudomonas aeruginosa is known to cause infections, particularly in individuals with certain risk factors such as intravenous drug use or having prosthetic valves.