Let's simplify the given expression step by step:

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

### Step 1: Factor the denominator of the first fraction
Notice that the denominator of the first fraction \(x^2 - 4\) can be factored as a difference of squares:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

So, the expression becomes:

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

### Step 2: Find a common denominator
To subtract the fractions, we need a common denominator. The least common denominator (LCD) is \((x - 2)(x + 2)\).

Rewrite the second fraction with this common denominator:

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

Now, the expression becomes:

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

### Step 3: Combine the fractions
Since both fractions now have the same denominator, we can subtract the numerators:

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

### Step 4: Expand and simplify the numerator
Let's expand the expression in the numerator:

\[
x^2 - \left[(x+1)(x-2)\right]
\]

Expanding \((x+1)(x-2)\) gives:

\[
x^2 - 2x + x - 2 = x^2 - x - 2
\]

So the numerator becomes:

\[
x^2 - (x^2 - x - 2)
\]

Simplify by distributing the negative sign:

\[
x^2 - x^2 + x + 2 = x + 2
\]

### Final Simplified Expression
The expression simplifies to:

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

Finally, cancel the common factor \((x + 2)\) in the numerator and denominator (assuming \(x \neq -2\) to avoid division by zero):

\[
\frac{1}{x - 2}
\]

This is the simplified form of the expression, valid for \(x \neq 2\) and \(x \neq -2\).

Would you like more details or have any questions? Here are some related questions:

1. What is the significance of the domain in this simplification?
2. How does factoring help in simplifying algebraic expressions?
3. Can you solve a similar expression with different denominators?
4. How would this process change if \(x^2 + 4\) was in the denominator instead of \(x^2 - 4\)?
5. What happens if \(x\) is equal to the values that make the original denominators zero?

**Tip:** Always check for factors that can be canceled only if they don't make the denominator zero after cancellation.

This page provides a detailed step-by-step guide on simplifying the rational expression (x^2/x^2 - 4) - (x+1/x+2). Key algebraic concepts include factoring the difference of squares and finding a common denominator. Suitable for students in Grades 9-10.

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