Bury the hatchet: end a quarrel or conflict and become friendly.
Bark up the wrong tree: to be wrong about the reason for something or the way to achieve something.
Blow off steam: to do or say something that helps you to get rid of strong feelings or energy.
Burn the midnight oil: read or work late into the night.
The sentence discusses two politicians who despite some differences make a certain decision to work together for the greater good. 'Bury the hatchet' fits in the context of the sentence.
The complete sentence is: Despite their differences, the two politicians decided to bury the hatchet and work together for the greater good.
Hence, option (a) is the answer.
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