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In English, adverbs of manner (e.g., quickly) are typically placed after the verb or the verb phrase.
Option 1 ("He quickly finished his homework for playing outside") is incorrect because "for playing" is an inappropriate construction here. The correct expression is "to play outside," indicating purpose.
Option 2 ("He finished quickly his homework to play outside") is awkward and ungrammatical. The adverb "quickly" should follow the verb phrase for natural flow.
Option 3 ("Quickly he finished his homework to play outside") sounds poetic or emphatic, but it’s not suitable for everyday use.
Option 4 ("He finished his homework quickly to play outside") correctly places the adverb "quickly" after the verb phrase and uses "to play" to express purpose. This makes it the most appropriate and grammatically sound choice.
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