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I'm very confused. Let me give you the sentence. I am making this sentence passive.

Sam and Dan _________________________invented XYZ. (fill in the blank with a form of "believe")

So I thought the correct format is:

Sam and Dan are believed to invented the XYZ.

Why is it "Sam and Dan are believed to have invented the XYZ" ?

If I change the sentence to using 'it' first it is like this:

It is believed that Sam and Dan invented XYZ.

In this sentence I don't need 'have' but the other one I need 'have'. I think I understand why (because invent is with an 'ed').

Why do I use the verb 'have'? Why not use a different verb? and how do I know when to use 'have' and when I don't need to use it.

1

Whenever we use an infinitival phrase to talk about the past, we normally have to use the present perfect:

King Henry VIII is thought (by scientists) to have had a rare genetic disorder that resulted in a series of miscarriages for two of his wives.

If we separate the infinitival phrase so that it be in the form of a that-clause, which can only be done by rewording the sentence a wee bit, we can see that the infinitival phrase is, indeed, in the past tense:

Scientists think that King Henry VIII had a rare genetic disorder that resulted in a series of miscarriages for two of his wives.

If we should choose not to use the present perfect in the infinitival phrase, the meaning is in the present tense; therefore, if I were to say,

King Henry VIII is thought (by scientists) to have a rare genetic disorder that has resulted in a series of miscarriages for two of his wives.

I would be, in essence, saying the equivalent of,

Scientists think that King Henry VIII has a rare genetic disorder that has resulted in a series of miscarriages for two of his wives.

That sentence above is clearly not talking about something in the past. Now, if I were talking about two past events occurring at the same time, the present perfect in the infinitival phrase wouldn't be correct:

He pretended to be a doctor. = He pretended that he was a doctor.

However, if the infinitival phrase is used to talk about an even more distant past event than the past event in the main clause, the present perfect is once again used:

Washington, D.C. was once thought to have been built over an ancient Indian burial ground. = People once thought that Washington, D.C. had been built over an ancient Indian burial ground.

I hope that might have helped you out. Take care and good luck!

1

"to [finite verb]" is never allowed in English, so "are believed to invented the XYZ" is wrong. Instead, "are believed to" must be followed by an infinitive verb.

"invented" can be the simple past tense or the past participle. So one solution is to form the perfect infinitive, "have invented".

The original question is questionable because "are believed to have" isn't really a form of "believe". But it's still the best answer available. Here are some other acceptable sentences using "are believed to [infinitive verb]":

Sam and Dan are believed to invent the XYZ. (changing "invent" to infinitive)

Sam and Dan are believed to have invented the XYZ.

Sam and Dan are believed to be the people who invented the XYZ.

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"are believed to have invented" is correct because the sentence says "Sam and Dan are believed to have something done". In this case "something done" is replaced with "invented the XYZ". It indicates that a person or a group of people believe that they have done something, which is inventing XYZ.

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