Correction Help

Meaning
Whereas infinitives express a present or future potential, perfect infinitives express a past potential. Compare:

(A) I will go with you.
(B) I would have gone with you.

In the first sentence (A), there is still a potential to go together. In the second sentence (B), the potential has passed and going together is no longer an option. The perfect infinitive is useful, therefore, to imagine a different past or express regret about a lost opportunity. For example:

(A) I could have caught that bus if I had run faster. We could have had it all. (Adele)

Form
The perfect infinitive is used with two forms: either 1) to + have + past participle, or 2) modal + have +past participle. Here are examples of the two forms.

The bomb blast appeared to have defeated the rebels, so the army was surprised by the second attack. In retrospect, they should have expected it.

Uses
Here are some uses of modals with perfect infinitives.
MUST + perfect infinitive: expresses a deduction or a logical conclusion:

You must have known that it was illegal.

CAN´T/COULDN´T + perfect infinitive: expresses impossibility or disbelief:

They can't have noticed yet. The office doesn't open until tomorrow.
You couldn't have been so foolish to think you would get away with it.

COULD + perfect infinitive expressed a past ability:

You could have passed if you had studied.

MAY/MIGHT + perfect infinitive expresses a past (strong or weak) possibility

They may have gotten lost.(strong)
They might have been attacked by bears. (weak)

SHOULD/SHOULDN'T + perfect infinitive expresses that a past obligation was not performed:

You should have called before coming here.
You shouldn't have dropped by so late.

WILL+ perfect infinitive expresses a time in the future when an event is already finished:

By the time you wake up, I will have returned.

WOULD + perfect infinitive is used in 3rd conditional structures for imagining a different past:

I would have told you the truth you if you had asked me.