The given expression is:

\[
\frac{1}{x} \cdot (2 + x) - \ln(x) \cdot x
\]

To simplify this expression, let's break it down step by step:

### Step 1: Simplify the first term
The first part of the expression is \(\frac{1}{x} \cdot (2 + x)\), which can be expanded as:

\[
\frac{1}{x} \cdot (2 + x) = \frac{2}{x} + \frac{x}{x} = \frac{2}{x} + 1
\]

### Step 2: Simplify the second term
The second part of the expression is \(- \ln(x) \cdot x\), which simplifies to:

\[
- \ln(x) \cdot x = - x \ln(x)
\]

### Step 3: Combine the terms
Now, combine both parts:

\[
\frac{2}{x} + 1 - x \ln(x)
\]

This is the simplified version of the expression:

\[
\frac{2}{x} + 1 - x \ln(x)
\]

### Final Expression:
\[
\boxed{\frac{2}{x} + 1 - x \ln(x)}
\]

Would you like further details or have any questions?

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Here are 5 related questions to expand on the topic:

1. How would you find the derivative of this expression?
2. What would the expression look like if you were to integrate it?
3. How does this expression behave as \( x \to \infty \)?
4. What is the domain of the simplified expression?
5. How would you solve the equation \(\frac{2}{x} + 1 - x \ln(x) = 0\)?

**Tip:** When simplifying logarithmic expressions, pay special attention to domain restrictions (e.g., \(\ln(x)\) is only defined for \(x > 0\)).

Learn how to simplify the expression 1/x * (2 + x) - ln(x) * x with a detailed breakdown of each step. Explore key algebraic and logarithmic concepts, including the expansion of rational expressions and natural logarithms. This problem is suitable for students in Grades 10-12.

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