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1. She ran off with the milkman and left her child with her husband? How could she abandon her own child?

2. She's thinking of running off with the milkman and leaving her child with her husband? How could she abandon her own child?

Q1) I think that in situation 1, "could" is used as a past tense, but in situation 1, "could" seems to imply "possibility" and I don't think in situation 2, "could" is used as a past tense. Am I right?

Q2) Is "could" in both sentences grammatically correctly used?

Thanks a lot for your help!

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fire1Q2) Is "could" in both sentences grammatically correctly used?

Yes.

Could is present tense in both. It expresses a critical reaction (horror) at this situation. It is relatively common in conversations.

The following is a quote from this source:

https://dictionary.cambridge.org/grammar/british-grammar/could

We often use the expression how could you/she/he/they? to show disapproval (to show that we don’t like what someone has done):

Grandfather, how could you? How could you leave me?

How could you have gone without telling me?

A:

We had to give away our dog when we moved to England.

B:

Oh, how could you?

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fire1I think that in situation 1, "could" is used as a past tense

As the sentence in question is a sort of linguistic formula, I'm inclined to agree with A.S.'s view that both 1 and 2 involve 'could' in present time. From that point of view it takes How could she have abandoned her own child? to put the remark into past time.

Nevertheless, you're right that 'could' can also express past time.

could — real past time ~ was able to
My nephew could speak several languages before he was ten years old.
OR
could — hypothetical present time ~ would be able to
You could be an engineer if you knew more mathematics.

CJ