If you want to read more about this structure, it is called the subjunctive. Basically verbs like suggest, recommend, insist, demand, etc. are followed by the verb in the present subjunctive mood. This is identical to the bare infinitive (the infinitive without to). This structure is little used in British English (and, as Dan says, can sound quite pompous). I think it is used more in American English. A British English grammar book will only discuss it briefly in the advanced level book. An American English grammar book might have more detail.

Correct answer: A. She suggested that he take a break. This is the correct sentence because after verbs like “suggest,” we use the base verb — that means the verb in its simple form, with no “to” and no -s, -ed, or -ing at the end. For example, the base form of: takes is take goes is go studied is study eating is eat So in this sentence, we say: "She suggested that he take a break." (base verb: take) This structure is used when we talk about suggestions, wishes, or important actions. It's called the subjunctive, but your main job is just to remember the pattern: suggest + that + someone + base verb Why the others are wrong: B. She suggested that he takes a break Wrong because “takes” is not the base verb — it has an -s. C. She suggested he to take a break Wrong because we don’t use “to” after “suggest someone.” D. She suggested he taking a break Wrong because “taking” is the -ing form, not the base verb. Summary: A base verb is the verb in its simplest form: no to, no -s, no -ed, no -ing Use this form after “suggest that” "She suggested that he take a break."

The academic answer is: only A is correct because after "suggest", we use the base form of the verb (not "takes" or "taking"); and C is wrong because "to" is unnecessary (either remove "to"; or change "he to" into "for him to"). But in reality, a lot of native speakers will say B, especially informally. A is a bit posh - if you speak like this around gangsters of inner-city streets, they will think you're fronting and may pick a fight with you. It's considered the proper way of speaking in a formal tone, but this is a nuance that a lot of less educated native speakers don't know, so I wouldn't recommend worrying about it too much. Most native speakers only know it by 'feeling' it after a lot of experience of hearing this kind of sentence in formal contexts.

Because the verb like suggestions, insist, demand are follow only base verb. So the suggested is followed by take.