She told me....... quiet because the baby was sleeping. (to be_to have been_that I was_that I am).                     I think the maker of the question needs the perfect infintive. What about the bare simple one? 

Original Post

It seems to me the only answer that fits is "to be", the bare simple form. It all happened in a past context without a present point of view.

A present point of view of looking back can justify the "to + have + past participle" pattern. For example, it's so good to have been in love

Hi, Kinto,

I agree that the simple infinitive is the correct form. This is only to clarify that we use the term "bare infinitive" to refer to the infinitive without "to," as in:

- He made me be quiet.

"that I was" would also be a possible answer in a special context. Suppose a child asks his/her mom:

- Why am I quiet?

and the answer is:

- You are quiet because the baby is sleeping.

The child may then report the answer as follows:

- She told me that I was quiet because the baby was sleeping.

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