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To determine the excluded values for the product \(\frac{x^2 - 4x - 21}{3x^2 + 6x} \cdot \frac{x^2 + 8x}{x^2 + 11x + 24}\), we need to identify the values of \(x\) that make any of the denominators equal to zero. 

Let's start with the individual denominators:

1. \(3x^2 + 6x\)
2. \(x^2 + 11x + 24\)

### Step 1: Factorize the Denominators

#### Factorizing \(3x^2 + 6x\):
\[ 3x^2 + 6x = 3x(x + 2) \]

#### Factorizing \(x^2 + 11x + 24\):
\[ x^2 + 11x + 24 = (x + 3)(x + 8) \]

### Step 2: Determine the Roots of the Denominators

Set each factor equal to zero to find the roots:

1. \(3x(x + 2) = 0\):
   - \(3x = 0 \Rightarrow x = 0\)
   - \(x + 2 = 0 \Rightarrow x = -2\)

2. \((x + 3)(x + 8) = 0\):
   - \(x + 3 = 0 \Rightarrow x = -3\)
   - \(x + 8 = 0 \Rightarrow x = -8\)

### Step 3: List All Excluded Values

The excluded values are the roots of the factored denominators:
\[ x = 0, -2, -3, -8 \]

These values will make at least one of the denominators zero, causing the entire expression to be undefined.

### Answer

The excluded values for the product are:
- \(-8\)
- \(0\)
- \(-3\)

The correct answers to select are:
- \(-8\)
- \(0\)
- \(-3\)

Would you like more details or have any questions?

### Additional Questions:
1. How do you factorize quadratic expressions?
2. What is the importance of excluded values in rational functions?
3. Can you explain why setting the denominator to zero identifies excluded values?
4. How would the solution change if there were more complicated expressions in the numerator?
5. What if one of the denominators was already factorized; how would you proceed?
6. How do you find the roots of a polynomial?
7. How do excluded values affect the graph of a rational function?
8. Can a rational function have more than one excluded value?

### Tip
Always check all parts of a rational expression (both numerators and denominators) for excluded values to ensure the function is defined throughout its domain.

Learn how to identify excluded values in rational functions using a detailed step-by-step guide. This problem covers key concepts such as rational functions, factoring quadratic expressions in denominators, and determining excluded values. Suitable for high school students studying algebra.

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