Let's simplify the expression step by step:

Given expression: $\frac{x^2 - x - 6}{x^2 - 9} + \frac{x^2 + 2x + 24}{x^2 - x - 12}$

Step 1: Factor the denominators

We start by factoring the quadratic expressions in the denominators.

1. Factor $x^2 - 9$: This is a difference of squares: $x^2 - 9 = (x - 3)(x + 3)$

2. Factor $x^2 - x - 12$: We need to find two numbers that multiply to $-12$ and add to $-1$. These numbers are $-4$ and $3$, so we can factor this as: $x^2 - x - 12 = (x - 4)(x + 3)$

Step 2: Rewrite the expression with factored denominators

Now, the expression becomes: $\frac{x^2 - x - 6}{(x - 3)(x + 3)} + \frac{x^2 + 2x + 24}{(x - 4)(x + 3)}$

Step 3: Simplify the numerators

Let’s attempt to factor the numerators.

1. Factor $x^2 - x - 6$: We need two numbers that multiply to $-6$ and add to $-1$. These numbers are $-3$ and $2$, so we can factor this as: $x^2 - x - 6 = (x - 3)(x + 2)$

2. Factor $x^2 + 2x + 24$: We need to see if this trinomial can be factored. However, the numbers don't seem to factor nicely. We will leave it as it is for now.

Step 4: Substitute the factored forms into the expression

Now, the expression becomes: $\frac{(x - 3)(x + 2)}{(x - 3)(x + 3)} + \frac{x^2 + 2x + 24}{(x - 4)(x + 3)}$

Step 5: Cancel out common factors

In the first term, $(x - 3)$ appears in both the numerator and denominator, so we can cancel it out: $\frac{x + 2}{x + 3} + \frac{x^2 + 2x + 24}{(x - 4)(x + 3)}$

Step 6: Combine the fractions

Now, both fractions have the same denominator $(x + 3)$, so we can combine them into a single fraction.

The combined fraction will be: $\frac{(x + 2)(x - 4) + (x^2 + 2x + 24)}{(x + 3)(x - 4)}$

Step 7: Simplify the numerator

Now, let's expand the terms in the numerator: $(x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8$ So the numerator becomes: $x^2 - 2x - 8 + x^2 + 2x + 24 = 2x^2 + 16$

Step 8: Final expression

Now, substitute this back into the fraction: $\frac{2x^2 + 16}{(x + 3)(x - 4)}$

Step 9: Factor out the common factor in the numerator

Notice that we can factor out a 2 from the numerator: $\frac{2(x^2 + 8)}{(x + 3)(x - 4)}$

Final Simplified Expression:

$\frac{2(x^2 + 8)}{(x + 3)(x - 4)}$

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Would you like more details on any step or have any questions?

Here are some related questions you might find useful:

1. How to factor a difference of squares?
2. How to factor quadratic trinomials?
3. What are the rules for adding fractions with different denominators?
4. How do you cancel common factors in algebraic fractions?
5. How to simplify a rational expression completely?

Tip: Always look for common factors in the numerator and denominator to simplify fractions before combining them!