((x-2)divided by x) plus (1 divided by x+1) divided by (x divided by x+2)

To solve the expression 

\[
\left(\frac{x-2}{x} + \frac{1}{x+1}\right) \div \left(\frac{x}{x+2}\right),
\]

let's follow the steps:

### Step 1: Simplify the Expression
First, express the division as multiplication by the reciprocal:

\[
\left(\frac{x-2}{x} + \frac{1}{x+1}\right) \times \frac{x+2}{x}.
\]

### Step 2: Distribute the Multiplication
Distribute \(\frac{x+2}{x}\) across each term inside the parentheses:

\[
\frac{x-2}{x} \times \frac{x+2}{x} + \frac{1}{x+1} \times \frac{x+2}{x}.
\]

### Step 3: Simplify Each Term
- For the first term: \(\frac{(x-2)(x+2)}{x^2}\).
  
  Use the difference of squares:

  \[
  \frac{x^2 - 4}{x^2}.
  \]

- For the second term:

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

So the entire expression becomes:

\[
\frac{x^2 - 4}{x^2} + \frac{x+2}{x(x+1)}.
\]

### Final Simplified Form
\[
\frac{x^2 - 4}{x^2} + \frac{x+2}{x(x+1)}.
\]

### Further Simplification
Since \(\frac{x^2 - 4}{x^2}\) can be expressed as \(1 - \frac{4}{x^2}\), the expression simplifies to:

\[
1 - \frac{4}{x^2} + \frac{x+2}{x(x+1)}.
\]

This is the simplified form of the given expression.

Would you like more details or have any questions?

### Relative Questions
1. How can you apply the distributive property to similar expressions?
2. What is the difference of squares, and how is it used in this problem?
3. How do you simplify complex fractions involving polynomials?
4. Can you explain how to find a common denominator for adding fractions?
5. What are some common mistakes to avoid when simplifying expressions like this?

### Tip
Always remember to factorize where possible to simplify expressions and look for common patterns like the difference of squares.

Learn how to simplify the fractional expression ((x-2)/x + 1/(x+1)) / (x/(x+2)) step-by-step. This problem involves algebraic concepts such as fractional expressions and polynomials. Suitable for students in Grades 10-12.

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