Final answer:
The most difficult and least productive approach to discovering exoplanets is direct imaging because exoplanets are faint and usually hidden by their star's glare.
Step-by-step explanation:
The approach to discovering exoplanets that has been the most difficult and least productive is direct imaging. This method is challenging due to exoplanets being faint and often obscured by the brilliant glare of the stars they orbit, making them incredibly hard to detect. The indirect methods, which include the transit method and Doppler shift method, have been more fruitful. Observing the effects of the planet on their star, rather than seeing the planet itself, has proven more effective. Large-scale application of the transit technique particularly benefited from the NASA Kepler space observatory, which successfully monitored thousands of stars for transits. On the other hand, the Doppler shift method has improved its ability to detect smaller planets as technology and observational time frames have expanded.