<s>[INST] How to solve this equation $(x-1)(x+2) + 4(x-1)\sqrt{\dfrac{x+2}{x-1}} = 12$? $(x-1)(x+2) + 4(x-1)\sqrt{\dfrac{x+2}{x-1}} = 12$
$Domain: x\in (-\infty;-2]\cup(1;+\infty)$
$x = 2\longrightarrow$ Done
How to prove $x = \dfrac{-1-3\sqrt{17}}{2}$ is the last solution? With raise both side by power of two?
Any better way such as factoring, grouping,...?
Please help!!
 [/INST] Assuming $x\neq 1$ you can rewrite your equation as
$$(x-1)(x+2) + 4\sqrt{(x-1)(x+2)} = 12$$
Then call $z = \sqrt{(x+1)(x-2)}$ and solve the quadratic associated equation, and then you pullback to $x$.
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