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The problem asks to determine the horizontal and vertical asymptotes of the curve given by:

\[
y = \frac{x^3}{x^2 + 3x - 10}
\]

### 1. **Vertical Asymptotes**:
Vertical asymptotes occur when the denominator of the function equals zero (since division by zero is undefined), provided the numerator doesn't also equal zero at the same points (which would create a hole instead of an asymptote).

The denominator is:

\[
x^2 + 3x - 10 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

So, the solutions are:

\[
x = \frac{-3 + 7}{2} = 2 \quad \text{and} \quad x = \frac{-3 - 7}{2} = -5
\]

Thus, the vertical asymptotes occur at **\( x = 2 \)** and **\( x = -5 \)**.

### 2. **Horizontal Asymptotes**:
To find the horizontal asymptote, we analyze the behavior of the function as \( x \to \infty \) or \( x \to -\infty \).

The degrees of the numerator and the denominator are 3 and 2, respectively (since the numerator has \( x^3 \) and the denominator has \( x^2 \)).

When the degree of the numerator is greater than the degree of the denominator (in this case, \( 3 > 2 \)), there is no horizontal asymptote. Instead, the curve will behave like a slant asymptote (or oblique asymptote) as \( x \to \infty \). To determine this, we can perform polynomial long division, but for now, it’s enough to conclude that there is **no horizontal asymptote**.

### Summary:
- **Vertical asymptotes**: \( x = 2 \) and \( x = -5 \)
- **No horizontal asymptote**, but potentially a slant asymptote.

Would you like a detailed explanation of the slant asymptote or any other clarifications?

### Follow-up Questions:
1. How do we find slant asymptotes from a rational function?
2. Why do vertical asymptotes occur at specific values of \( x \)?
3. What happens at a hole in the graph of a rational function?
4. How does polynomial long division help in identifying slant asymptotes?
5. Can a rational function have both vertical and horizontal asymptotes?

### Tip:
For determining horizontal asymptotes, comparing the degrees of the numerator and denominator gives you a quick clue about their existence.

Determine the horizontal and vertical asymptotes of the curve y = x^3 / (x^2 + 3x - 10).

Learn how to determine the horizontal and vertical asymptotes of the rational function y = x^3 / (x^2 + 3x - 10). This problem covers key algebraic concepts such as solving quadratic equations and analyzing asymptotic behavior of functions. Suitable for students in Grades 11-12.

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