has - verb
trouble - direct object of verb
coping - ? (gerund modifier for trouble?)
coping with school - ? (is it a verbal phrase that's using a gerund/phrasal verb which, in turn, acts as modifier for trouble?)
if it were 'she has coping trouble',
would coping = gerund modifier of trouble?
NOT A TEACHER
(1) Congratulations! In my opinion as a non-teacher, I believe that
you are 100% correct:
(a) "Coping with school" is a gerund phrase.
(b) The gerund phrase has some connection to "trouble."
(2) I have checked the Web and my books, and I am delighted to
share what I found:
Theory No, l
Some people believe that your sentence has a suppressed (not used)
preposition:
She had trouble
in coping with school.
It seems that most native speakers do not say/write the preposition, but
it is necessary if your teacher asks you to explain why "coping" is a
gerund.
Theory No. 2
Some people prefer to say that "coping with school" is a gerund
phrase acting as an objective complement.
If I say, " I painted my house," a friend might ask, "What color?"
I would answer: "I painted my house
red." As you can see, the
word "red" refers to the object ("house") and completes the meaning
of the words ("I painted my house."). So books call this an
objective
complement.
Thus, if you say that "She has trouble," probably someone would
ask you about the kind of trouble. And you might answer, "Oh,
I'm sorry for not being more complete. She is having trouble
coping with school. ("Coping with school" would be the objective
complement of "trouble.")
(3) It would very interesting to know which theory your teacher
supports. In fact, s/he may have another theory or explanation.
Thank you for your great question. It forced me to review my
own understanding.
Sincerely yours,
James