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.