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the student's solution is not the best way to go about solving the problem
instead of multiplying both sides by (2x^2+x-15)(2x-5)(x+3), which is messy to work with,
he should have recognized that, on the left side, he can factor the denominator into (2x-5)(x+3), and on the right side, he can rewrite both fractions such that their denominators are (2x-5)*(x+3).
the "best" thing for the student to do, is to do what they feel the most comfortable with.
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In order to solve a problem like this, one method is to multiply both sides by the LCD of all the fractions. Since the left denominator factors into (2x - 5)(x + 3), the LCD of all fractions is simply (2x - 5)(x + 3). All he needed to do is to multiply by the LCD instead of the longer expression he used. By doing what he did, he made the problem longer to solve than it had to be.
but, if you dont know what the LCD is to start with, any CD will work ...