Answer:
A. Walking along the shore, Rosie and her little sister began to notice many starfish on the rocks.
Explanation:
Participial phrase is a set of words whose main part is participle and as a whole functions as an adjective.
Infinitive phrase has an infinitive as main part and can have many functions in a sentence.
Sentences C. and D. don't have infinitive at all (to + verb).
Sentence B. has infinitive (to calm down) but it doesn't havs participle. "taking" although looks like participle, is actually gerund. The whole phrase (taking a long walk) doesn't modify any noun, it functions as a subject.
Finally, sentence A. has an infinitive (to notice) and infinitive phrase (to notice many starfish on the rocks). Since this modifies verb "began" this server as an adverb. Also, there iss a participle (walking) and participial phrase (walking along the shore). This functions as an adjective because it modifies Rosa and her sister.
