\frac{\left(\frac{x^2+2}{x-3}\right)}{\left(1+\sqrt{x}\right)}

The given expression is:

\[
\frac{\left(\frac{x^2+2}{x-3}\right)}{\left(1+\sqrt{x}\right)}
\]

To simplify this expression, we can treat it as a compound fraction (or complex fraction) and apply the following steps:

1. Rewrite the expression:
   \[
   \frac{x^2 + 2}{(x - 3)(1 + \sqrt{x})}
   \]
   This is obtained by multiplying the denominator of the outer fraction with the denominator of the inner fraction.

So, the simplified form of the given expression is:

\[
\frac{x^2 + 2}{(x - 3)(1 + \sqrt{x})}
\]

This is the most simplified form of the expression unless there are specific restrictions or values for \(x\) that need to be considered (such as \(x \geq 0\) for the square root to be valid, and \(x \neq 3\) for division by zero).

Would you like more details on how this was simplified or have any further questions? Here are 5 related questions you might want to explore:

1. How can we evaluate this expression for a specific value of \(x\)?
2. What are the restrictions on \(x\) for this expression to be defined?
3. How can we handle the square root term more effectively in simplifications?
4. How would you differentiate or integrate this expression?
5. What happens to the expression as \(x\) approaches certain values, such as \(3\) or \(0\)?

**Tip:** When simplifying complex fractions, always try to rewrite them as a single fraction to avoid confusion.

This webpage provides a step-by-step simplification of a complex fraction involving algebraic expressions and square roots: \frac{\left(\frac{x^2+2}{x-3}\right)}{\left(1+\sqrt{x}\right)}. Learn how to combine fractions and simplify using algebraic rules. Suitable for students in Grades 9-12.

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