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The problem asks to find the domain of the function \( k(x) = \frac{x^2 + 5x + 1}{x^3 - x^2 - 6x} \) using interval notation.

### Steps to find the domain:
The domain of a rational function is determined by the values of \( x \) for which the denominator is not zero, since division by zero is undefined.

1. **Denominator:** Set the denominator equal to zero and solve for \( x \):
   \[
   x^3 - x^2 - 6x = 0
   \]

2. **Factor the denominator:**
   Factor out \( x \):
   \[
   x(x^2 - x - 6) = 0
   \]

3. **Factor the quadratic expression:**
   \[
   x^2 - x - 6 = (x - 3)(x + 2)
   \]
   So, the denominator becomes:
   \[
   x(x - 3)(x + 2) = 0
   \]

4. **Find the values of \( x \):**
   The denominator equals zero when \( x = 0 \), \( x = 3 \), or \( x = -2 \).

   Therefore, the function is undefined at these values.

### Domain:
The domain is all real numbers except \( x = 0 \), \( x = 3 \), and \( x = -2 \). In interval notation, this is:
\[
(-\infty, -2) \cup (-2, 0) \cup (0, 3) \cup (3, \infty)
\]

Let me know if you'd like more detailed explanations or have any further questions!

#### Related questions:
1. How do you determine the domain of a function in general?
2. What are the steps to factor a quadratic expression?
3. Why is division by zero undefined in mathematics?
4. Can the domain of any function include infinity?
5. How would the domain change if the numerator was also factorable?

#### Tip:
Always check for any restrictions in both the denominator and numerator to identify all possible exclusions from the domain!

Give the domain of k(x) = (x^2 + 5x + 1) / (x^3 - x^2 - 6x) using interval notation.

Learn how to find the domain of the rational function k(x) = (x^2 + 5x + 1) / (x^3 - x^2 - 6x). This problem involves factoring the denominator to identify excluded values, using algebraic methods to determine the domain in interval notation. Suitable for Grades 9-12.

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