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Gabby: Susan, are you okay?

Susan: I'm just so angry. If Mary Alice was having problems, she should have come to us. She should have let us help her.

Gabby: What kind of problems could she have had? She was healthy, had a great home, a nice family. No, if Mary Alice was having some sort of crisis, we'd have known. She lives 50 feet away, for god's sake.

Susan: Gabby, the woman killed herself. Something must have been going on.

TV Series: Desperate Housewives

Why Gabby didn't use "What kind of problems could she have?" because "could" can be used to express past possibility!

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"could" can be used to express present or past possibility, depending on the context:

  • She looks worried. She could have some problem. (Also: She could be having some problem.)

  • She looked worried. She could have some problem. (Also: She could be having some problem.)

When the past is hypothetical and refers to a possibility that DID NOT come true, then could is followed by a perfect infinitive:

  • She could have had some problem (if she had gone there).

However, in the text "could" is followed by a perfect infinitive because the woman is dead, and the possibility cannot be confirmed. Actually, a deduction is being made about the past, just as is the case with "must have":

  • What kind of problems could she have had?

  • Something must have been going on.

Notice that all the modals in the text are followed by perfect infinitive, showing that reference is being made to something in the past:

  • If Mary Alice was having problems, she should have come to us. She should have let us help her.
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When you are talking about something you possess in some way, the phrase "have had" is basically the past tense of "have".

I have an apple

Means I have an apple now.

I have had an apple

Means I have had an apple in the past.

Problems are a thing it is possible for you to have. Therefore,

What kind of problems could she have?

Means "What kind of problems is it possible for her to have at the moment?"

Whereas

What kind of problem could she have had?

Means "What kind of problems is it possible for her to have had in the past?"

In your previous highlighted sentence, "have" is being used in a different way.

She should have let us help her

"She" does not "have" anything, rather, she "should have" done something in the past. The word "have" is being used in a different way, so it would not be correct to say

She should have had let us help her

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