To simplify the given rational expression:

$\frac{x^2 + 7x - 8}{x^2 + 5x - 24}$

we need to factor both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $x^2 + 7x - 8$. We are looking for two numbers that multiply to $-8$ and add to $7$.

These numbers are $8$ and $-1$. So, we can factor the numerator as:

$x^2 + 7x - 8 = (x + 8)(x - 1)$

Step 2: Factor the denominator

The denominator is $x^2 + 5x - 24$. We are looking for two numbers that multiply to $-24$ and add to $5$.

These numbers are $8$ and $-3$. So, we can factor the denominator as:

$x^2 + 5x - 24 = (x + 8)(x - 3)$

Step 3: Simplify the expression

Now, substitute the factored forms of the numerator and denominator:

$\frac{(x + 8)(x - 1)}{(x + 8)(x - 3)}$

We can cancel out the common factor $(x + 8)$, as long as $x \neq -8$ (to avoid division by zero):

$\frac{x - 1}{x - 3}, \quad x \neq -8$

Final simplified form:

$\frac{x - 1}{x - 3}, \quad x \neq -8$

Let me know if you would like more details or further clarifications!

Related questions:

1. How do you factor quadratic expressions in general?
2. What happens if the quadratic can't be factored using integers?
3. How do you handle rational expressions with higher degree polynomials?
4. What are the conditions for canceling factors in rational expressions?
5. How do you deal with undefined values in rational expressions?

Tip:

Always check for any restrictions (values that make the denominator zero) when simplifying rational expressions.