The given expression is:

$\frac{x+1}{x+2} + \frac{x-2}{x^2-4}$

First, notice that $x^2 - 4$ can be factored as:

$x^2 - 4 = (x+2)(x-2)$

So, the expression becomes:

$\frac{x+1}{x+2} + \frac{x-2}{(x+2)(x-2)}$

Now, both terms share $x+2$ as a common factor in the denominator. We rewrite the second term as:

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

Thus, the expression simplifies to:

$\frac{x+1}{x+2} + \frac{1}{x+2}$

Now, since the denominators are the same, we can combine the numerators:

$\frac{(x+1) + 1}{x+2} = \frac{x+2}{x+2}$

Finally, the expression simplifies to:

$1$

Final Answer: $1$

Would you like more details or have any other questions?

Here are five related questions to extend your understanding:

1. How do you factor quadratic expressions like $x^2 - 4$?
2. What is the process for adding rational expressions?
3. Can you explain the importance of common denominators when adding fractions?
4. How does factoring help simplify algebraic expressions?
5. What happens if the denominator equals zero in a rational expression?

Tip: Always look for factoring opportunities to simplify expressions before proceeding with addition or subtraction!