algebraic expression - square root issue

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Assuming $x\ge1$. How to go from $$ \frac{\sqrt {x-1}+\sqrt{x+1}}{\sqrt {x-1}-\sqrt{x+1}} $$ to $$ -\sqrt{x^2-1}-x $$ ?

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divide and multiply by $(\sqrt{x-1}+ \sqrt{x+1})$ to get the answer