Answer:
A. Determiner
Step-by-step explanation:
To answer this question we need to understand what determiner is.
A determiner is a word that introduces a noun. It always comes before a noun, not after; it can come before an adjective used to describe the noun.
There are 4 main types of determiners: articles, demonstratives, quantifiers, and possessives.
A cat is a good pet - determiner-article
These black raspberries are sour - determiner-demonstrative
He ate all the cookies - determiner-quantifier
It's our secret recipe - determiner-possessive
As for our sentence - These bagels are stale - the word 'these' is a determiner (demonstrative pronoun).
Demonstrative pronouns are also used as determiners in English. There are four of them: this, that, these and those. Demonstratives are used in a situation in which the speaker can point to the item they mean, making them even more specific than a definite article.
To sum up, the word 'these' is a demonstrative pronoun and we can think option B is correct as it is a pronoun; but exactly in this sentence, it is a pronoun that plays a role of a determiner.
I hope it helped you :)