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To find the domain of the rational expression given in the question, we need to determine which values of \( x \) make the expression undefined. The expression is:

\[
\frac{9x^2 - 13x - 10}{x(x^2 + 3x - 10)}
\]

For a rational expression, the domain includes all real values of \( x \) except those that make the denominator equal to zero.

### Step 1: Set the Denominator to Zero
The denominator is:
\[
x(x^2 + 3x - 10)
\]

For this expression to be undefined, we set the denominator equal to zero:
\[
x(x^2 + 3x - 10) = 0
\]

This equation will be zero if either \( x = 0 \) or \( x^2 + 3x - 10 = 0 \).

### Step 2: Solve \( x^2 + 3x - 10 = 0 \)
Now we solve the quadratic equation \( x^2 + 3x - 10 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = 3 \), and \( c = -10 \).

Substitute the values:
\[
x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-10)}}{2 \cdot 1}
\]
\[
x = \frac{-3 \pm \sqrt{9 + 40}}{2}
\]
\[
x = \frac{-3 \pm \sqrt{49}}{2}
\]
\[
x = \frac{-3 \pm 7}{2}
\]

This gives us two solutions:
1. \( x = \frac{-3 + 7}{2} = 2 \)
2. \( x = \frac{-3 - 7}{2} = -5 \)

### Step 3: Exclude Values from the Domain
The values of \( x \) that make the denominator zero are \( x = 0 \), \( x = 2 \), and \( x = -5 \). Therefore, these values must be excluded from the domain.

### Final Answer
The domain of the rational expression is all real numbers \( x \) such that:
\[
x \neq 0, \, x \neq 2, \, x \neq -5
\]

### Matching the Answer to the Options
From the provided options, the correct choice is:
- **D.** \( x \) is a real number and \( x \neq 0 \), \( x \neq -5 \), and \( x \neq 2 \).

Let me know if you would like more details or have any questions!

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Here are some additional questions you could consider:
1. What is the significance of finding the domain in rational expressions?
2. How does factoring help in simplifying rational expressions?
3. What are some other ways to solve quadratic equations besides the quadratic formula?
4. How does the domain of a function affect its graph?
5. Can the domain of a rational expression ever include complex numbers?

**Tip:** Always check if a factor in the denominator can be simplified with the numerator; this can sometimes reveal removable discontinuities.

Find the domain of the rational expression: (9x^2 - 13x - 10) / (x(x^2 + 3x - 10))

Learn how to determine the domain of a rational expression, specifically (9x^2 - 13x - 10) / (x(x^2 + 3x - 10)). Follow a step-by-step solution using the quadratic formula to find values of x that make the denominator zero. Suitable for high school students in Grades 9-12.

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