I am not completely sure, but I would say that the apothecary suggests that Jane go to school because he believes Jane would be happier and healthier away from Gateshead. She tells him how unhappy she is, so that is why he wants to help her.
Where the poem? Or the picture
The sentence that is an example of a faulty coordination is: The guitar is used in a lot of contemporary music, and the instrument probably originated in Spain during the 1500s. What makes this sentence's coordination faulty is that, the first part talks about guitars are used in contemporary music, but on the second clause, the topic became the history of the instrument. There is no coherence in the sentence.