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To find the domain of \( f(x) = \frac{r(x)}{q(x)} \), we need to determine when the function is undefined. A rational function like \( f(x) \) is undefined when the denominator \( q(x) \) is equal to zero. 

Here are the given functions:
- \( r(x) = (x + 2)(x - 4)(x + 5) \)
- \( q(x) = x^2 + 7x + 10 \)

Let's find the zeros of \( q(x) \) by solving \( q(x) = 0 \).

### Step 1: Factor \( q(x) \)
The function \( q(x) = x^2 + 7x + 10 \) can be factored as follows:
\[
q(x) = (x + 5)(x + 2)
\]

### Step 2: Set \( q(x) = 0 \) and solve for \( x \)
Setting each factor in \( q(x) = (x + 5)(x + 2) = 0 \) equal to zero:
\[
x + 5 = 0 \Rightarrow x = -5
\]
\[
x + 2 = 0 \Rightarrow x = -2
\]

So, \( q(x) = 0 \) when \( x = -5 \) or \( x = -2 \).

### Step 3: Determine the domain of \( f(x) \)
The domain of \( f(x) = \frac{r(x)}{q(x)} \) is all real numbers except for the values of \( x \) that make \( q(x) = 0 \). Thus, the domain of \( f(x) \) is:
\[
\text{Domain of } f(x) = \{ x \in \mathbb{R} \mid x \neq -5 \text{ and } x \neq -2 \}
\]

### Answer:
The domain of \( f(x) \) is all real numbers except \( x = -5 \) and \( x = -2 \).

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What would be the range of \( f(x) \) given these functions?
2. How do the zeros of \( r(x) \) affect the behavior of \( f(x) \)?
3. What happens to \( f(x) \) as \( x \) approaches -5 or -2?
4. How do you find the vertical asymptotes of a rational function like \( f(x) \)?
5. Can you identify any holes in the graph of \( f(x) \) if \( r(x) \) and \( q(x) \) share factors?

**Tip:** When determining the domain of a rational function, always look for values of \( x \) that make the denominator zero, as these values are excluded from the domain.

The function r is given by r(x) = (x + 2)(x - 4)(x + 5), and the function q is given by q(x) = x^2 + 7x + 10. Let f be the function given by f(x) = r(x) / q(x). What is the domain of f?

Learn how to find the domain of a rational function f(x) = r(x) / q(x), where r(x) = (x + 2)(x - 4)(x + 5) and q(x) = x^2 + 7x + 10. This solution covers essential algebraic concepts such as factoring polynomials and using the zero-product property. Suitable for students in Grades 10-12.

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