Answer:
The - article/adjective
Popularity - noun
Of baseball - prepositional phrase (of is a preposition, and baseball is a noun)
Grew - verb
Quickly - adverb
Throughout the country - prepositional phrase (throughout is a preposition, and country is a noun)
Step-by-step explanation:
Let's start by identifying the subject of the sentence.
The popularity of baseball grew quickly throughout the country.
Popularity is a noun.
Now, what was popularity doing? Answering this question gives us the verb.
The popularity...grew...
Grew is a verb.
"The" describes popularity, and is therefore considered an adjective. More specifically, "the" is an article. Articles include "the" and "a/an".
Notice that there is another word in the sentence that describes something else. This word is "quickly". However, since it describes a verb, it is know as an adverb.
Now, we are left with "of baseball", and "throughout the country". Both "of" and "throughout" are prepositions, which make "of baseball", and "throughout the country" prepositional phrases (always start with a preposition and end in a noun or pronoun). A quick trick to identifying prepositions is using the "log trick", where you add the preposition before "the log".
Ex: "Through the log"