Subjunctive past

In a sentence where the main proposition is in the present or future tense of the indicative, the past tense of the subjunctive in the subordinate is used to remain consistent with the concordance of the tenses and thus express an event before the one stated in the main.

Je ne pense pas qu'il ait terminé à temps.
Je regretterai que mes amis soient restés chez eux.

In order to form the past tense of the subjunctive, it is necessary to know how to conjugate the auxiliaries "avoir" and "être" in the present tense of the subjunctive and how to form the past participle. The main difficulty of the past tense of the subjunctive comes from the agreements with the past participle.

Here is an example of a verb conjugated in the past tense of the subjunctive with the auxiliaries "avoir" and "être":

Subject	Auxiliary (avoir)	Past participle (finir)
que j'	aie	fini
que tu	aies	fini
qu'il	ait	fini
que nous	ayons	fini
que vous	ayez	fini
qu'ils	aient	fini

Subject	Auxiliary (être)	Past participle (venir)
que je	sois	venu
que tu	sois	venu
qu'l	soit	venu
que nous	soyons	venus
que vous	soyez	venus
qu'ils	soient	venus

It is immediately apparent that the question of the agreement arises. With the auxiliary "être", the agreement is made with the subject from where the "s" came because the subject is put for several people. If it is a woman speaking, an "e" is also added to indicate the feminine: "que je sois venue".
The agreement is more delicate with the auxiliary "avoir" because it is necessary to make the agreement with the direct object complement when it is placed in front and not with the subject.

The past tense of the subjunctive is formed from the auxiliaries "avoir" and "être" conjugated to the present tense of the subjunctive to which the past participle is added.
With the auxiliary "être", we agree the past participle with the subject.
With the auxiliary "avoir", the agreement of the past participle is done with the direct object complement when it is placed in front of it.