The electrolysis of bauxite is conducted using the Hall-Heroult process. This industrial process involves dissolving alumina (Al₂O₃) from bauxite in molten cryolite and passing a direct electric current through the mixture. The process reduces aluminum ions to aluminum metal at the cathode and forms oxygen gas at the anode. This highly energy-intensive process is the primary method for aluminum extraction globally.

Other options, such as the Haber's process, Born-Haber cycle, and ion exchange method, are unrelated to the extraction of aluminum or the electrolysis of bauxite. The Haber's process is for ammonia synthesis, the Born-Haber cycle is a theoretical model for ionic compounds, and the ion exchange method is used for purifying or separating ions in solution.

The Haber's process is a method used to synthesize ammonia from nitrogen and hydrogen gases. It is not related to electrolysis or the extraction of aluminum from bauxite.

The Born-Haber cycle is a theoretical model used to analyze the energies involved in forming ionic compounds, particularly lattice energy. It does not involve electrolysis or any practical extraction of metals from ores.

The ion exchange method is used to separate ions in solutions, commonly in water treatment or purification processes. It is not a method used for electrolyzing bauxite to extract aluminum.

The Hall-Heroult process is the correct method for the electrolysis of bauxite. It involves the electrolytic reduction of alumina (Al₂O₃) dissolved in molten cryolite (Na₃AlF₆) to produce aluminum metal.