Question

Could you help me find my running shoes? running is in italics It is a participle functioning as a noun. It is a participle functioning as an adjective. It is an infinitive functioning as a noun. It is an infinitive functioning as an adjective.

Answer 1

A participle normally ends in -ing or -ed. 
An infinitive is the simplest stem word of something.

An example of these would be running and run. Running is a participle and run is an infinitive. 

Because you now know this, you can immediately eliminate anything that does not say participle, since we now know running is one. You can eliminate the last two choices. They are wrong.

Running can only ever be used as an adjective and a verb, never a noun. Adjectives are used to describe nouns and verbs are used to describe action that a subject takes.

Could you help me find my running shoes? 

In this sentence, the subject is not running. The shoes are not literally running either. 

Could you help me find my running shoes? 

Running describes the word "shoes" making it an adjective. The shoes are being described as a pair of footwear the speaker wears when they go running.

This means that this is a participle functioning as an adjective.