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The main problem here is that you're trying to reach an erroneous goal. The answer you have listed is wrong--the numerator should be u(u+v)^2.
Your work up to where you stopped is solid, so I'm not sure why you started having problems where you did. All there is left to do is make some cancelations. Remember that addition is commutative, so the (v+u) in the numerator of your right multiplicand can be changed around to (u+v) to match the (u+v) in the denominator of the left multiplicand. Now, start crossing things out.
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The Jamie Star Scenario
Ahh, the answer is wrong! That explains a lot, I've been scratching my head over that problem for ages. I see it now though, thanks!
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