Let's simplify the given expression:

$\frac{5x^2 - 5x - 60}{x^2 + 8x + 15}$

Step 1: Factor the numerator and denominator

Factor the numerator: $5x^2 - 5x - 60$

First, factor out the common factor of 5:

$5(x^2 - x - 12)$

Now, factor the quadratic $x^2 - x - 12$. We look for two numbers that multiply to -12 and add up to -1. These numbers are -4 and 3.

So, we factor it as:

$x^2 - x - 12 = (x - 4)(x + 3)$

Thus, the numerator becomes:

$5(x - 4)(x + 3)$

Factor the denominator: $x^2 + 8x + 15$

We look for two numbers that multiply to 15 and add up to 8. These numbers are 3 and 5.

So, the denominator factors as:

$x^2 + 8x + 15 = (x + 3)(x + 5)$

Step 2: Simplify the expression

Now substitute the factored forms of the numerator and denominator into the expression:

$\frac{5(x - 4)(x + 3)}{(x + 3)(x + 5)}$

We can cancel the common factor of $(x + 3)$ from the numerator and denominator:

$\frac{5(x - 4)}{x + 5}$

Final simplified expression:

$\frac{5(x - 4)}{x + 5}$

This is the simplified form of the given expression.

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Would you like to explore any further details, or is there another aspect of this problem you'd like help with?

Follow-up questions:

1. How do you factor quadratics with coefficients other than 1?
2. What happens if $x = -3$ in the original expression?
3. How can you check if your factorization is correct?
4. What are the restrictions on the domain of this rational expression?
5. What is the role of factoring in simplifying rational expressions?

Tip: Always look for common factors first when simplifying expressions. This can make factorization much easier!