The correct option is A most politely
Adverbs modify a verb, an adjective and sometimes another adverb in a sentence.
Some adverbs, like adjectives, have three degrees of comparison.
If the adverb is of one syllable, we form the comparative by adding er, and the superlative by adding est, to the positive; as, 'fast-faster-fastest'.
Adverbs ending in ly form the comparative by adding more and the superlative by adding most; as, 'swiftly-more swiftly-most swiftly'.
Here,
Option B is correct as most politely is the superlative form of politely. Consequently the other three options cancel out.