How are negative expressions (ne pas, ne aucun(e), ne que, ne jamais, ne plus) formed in the passé composé and is there a way to predict whether it forms around or after the auxilary?
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There is no such thing as "or after" the auxiliary. The negative of these forms is formed by wrapping the ne+pas/whatever circumfix around the auxiliary and any accompanying pronoun:
In most spoken forms of French (and writing that emulates them), the ne is almost universally dropped, which is the only way I can think of that one could describe a negative as being placed "after" the auxiliary.
in passé composé most of the adverbs come in between the auxiliary verb and the past participle.
NOTE: Adverbs of place and time come after the past participle
SOURCE: apprenons le français- méthode de français (4)