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The problem presented in the image involves simplifying and evaluating a mathematical expression \( Q \). The given expression is:

\[
Q = \left(\frac{1}{\sqrt{x-1}} - \frac{1}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x+1}} + \frac{2}{\sqrt{x-1}}\right)
\]

### Part (a): Simplify the Expression \( Q \)

To simplify \( Q \), let's expand the expression by multiplying the two terms together:

\[
Q = \left(\frac{1}{\sqrt{x-1}} - \frac{1}{\sqrt{x+1}}\right)\left(\frac{1}{\sqrt{x+1}} + \frac{2}{\sqrt{x-1}}\right)
\]

This can be expanded using the distributive property:

\[
Q = \frac{1}{\sqrt{x-1}} \cdot \frac{1}{\sqrt{x+1}} + \frac{1}{\sqrt{x-1}} \cdot \frac{2}{\sqrt{x-1}} - \frac{1}{\sqrt{x+1}} \cdot \frac{1}{\sqrt{x+1}} - \frac{1}{\sqrt{x+1}} \cdot \frac{2}{\sqrt{x-1}}
\]

Simplifying each term:

1. \(\frac{1}{\sqrt{x-1}} \cdot \frac{1}{\sqrt{x+1}} = \frac{1}{\sqrt{(x-1)(x+1)}} = \frac{1}{\sqrt{x^2-1}}\)
2. \(\frac{1}{\sqrt{x-1}} \cdot \frac{2}{\sqrt{x-1}} = \frac{2}{x-1}\)
3. \(\frac{1}{\sqrt{x+1}} \cdot \frac{1}{\sqrt{x+1}} = \frac{1}{x+1}\)
4. \(\frac{1

Explore how to simplify and evaluate a complex algebraic expression involving square roots. This problem demonstrates the use of the distributive property in algebraic expressions. Suitable for high school students studying algebra.

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