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To find the horizontal asymptote for the given rational function \( p(x) = \frac{4x^2 + 11x - 6}{2x^2 + 2x - 24} \), let's proceed step-by-step.

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### Horizontal Asymptote Rules:
1. **If the degrees of the numerator and denominator are the same**, the horizontal asymptote is \( y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} \).
2. **If the degree of the numerator is less than the degree of the denominator**, the horizontal asymptote is \( y = 0 \).
3. **If the degree of the numerator is greater than the degree of the denominator**, there is no horizontal asymptote (it may have an oblique asymptote instead).

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### Step-by-Step Solution:
The given function is:

\[
p(x) = \frac{4x^2 + 11x - 6}{2x^2 + 2x - 24}
\]

- The **degree** of the numerator (top) is 2 (since the highest power of \(x\) is \(x^2\)).
- The **degree** of the denominator (bottom) is also 2.

Since both the numerator and the denominator have the same degree, the horizontal asymptote is given by:

\[
y = \frac{\text{leading coefficient of the numerator}}{\text{leading coefficient of the denominator}}
\]

The leading coefficient of the numerator is **4**, and the leading coefficient of the denominator is **2**. So:

\[
y = \frac{4}{2} = 2
\]

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### Final Answer:
The horizontal asymptote is:

\[
y = 2
\]

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Would you like more details or have further questions?

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### Related Questions:
1. How do you find vertical asymptotes for rational functions?
2. What happens if the degree of the numerator is greater than the denominator?
3. Can a function have both horizontal and oblique asymptotes?
4. How do you determine the behavior of the function as \(x \to \infty\) beyond the asymptotes?
5. What is the difference between horizontal asymptotes and slant (oblique) asymptotes?

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### Tip:
To confirm the presence of horizontal asymptotes, always compare the degrees of the numerator and the denominator first—it simplifies the process.

Find the horizontal asymptote, if any, for the following function: p(x) = (4x^2 + 11x - 6) / (2x^2 + 2x - 24).

Learn how to find the horizontal asymptote of the rational function p(x) = (4x^2 + 11x - 6) / (2x^2 + 2x - 24). This solution covers key algebraic concepts like horizontal asymptotes and rational functions. Suitable for high school students in Grades 10-12.

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