Answer:
The words that would correctly complete the sentence are the following ones:
more quickly.
Explanation:
The sentence will be like this:
Do lentils cook more quickly than pinto beans?
The sentence is an example of comparative of superiority, since two kinds of food- lentils and pinto beans- are being compared to each other in the sense that one can be "more" than the other one in some aspect- here, the one that cooks more quickly.
In terms of structure, "more quickly" is the way to go, since quickly is an adverb of manner which has more than one syllable and because of that needs to be preceded by "more" in a comparative sentence.
"Most quickly" and "most quickest" do not make sense here, since both are in superlative form, not comparative. Besides, there are grammar inaccuracies there.
"More quicker", in spite of being in comparative form, is grammatically incorrect, since the adjective "quick" has a single syllable, and because of that would not need to be preceded by "more" in a comparative sentence, since the suffix "er" already does its job.
