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The problem you provided asks to determine the end behavior of the given rational function:

\[
f(x) = \frac{-6x^5 + 8x^4 + 3x}{6x^2 - 4x - 5}
\]

### Step 1: Analyze the degree of the numerator and denominator
- The **numerator** is \(-6x^5 + 8x^4 + 3x\), which has the highest degree term of \(-6x^5\). The degree of the numerator is 5.
- The **denominator** is \(6x^2 - 4x - 5\), which has the highest degree term of \(6x^2\). The degree of the denominator is 2.

### Step 2: Determine the end behavior
To find the end behavior of \(f(x)\) as \(x \to \infty\) and \(x \to -\infty\), we focus on the highest-degree terms in both the numerator and denominator.

For large values of \(x\), the function behaves like:

\[
f(x) \approx \frac{-6x^5}{6x^2} = -x^3
\]

This tells us that as \(x\) increases or decreases without bound, the function behaves like a cubic function (\(-x^3\)).

- As \(x \to \infty\), \(-x^3 \to -\infty\), so \(f(x) \to -\infty\).
- As \(x \to -\infty\), \(-x^3 \to \infty\), so \(f(x) \to \infty\).

### Conclusion:
- \(f(x) \to -\infty\) as \(x \to \infty\).
- \(f(x) \to \infty\) as \(x \to -\infty\).

Would you like further clarification on any part of this solution?

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Here are five related questions to expand on this:
1. How would the end behavior change if the degree of the denominator was higher than the numerator?
2. What happens to the end behavior if we reverse the signs of the leading terms in the numerator and denominator?
3. How do you determine horizontal asymptotes in rational functions?
4. Can you predict the behavior at \(x = 0\) for this function without graphing?
5. What effect do the lower-degree terms have on the function's behavior near the origin?

**Tip:** When analyzing rational functions, always focus on the highest-degree terms for determining end behavior.

Consider the following rational function: f(x) = (-6x^5 + 8x^4 + 3x) / (6x^2 - 4x - 5). Determine f’s end behavior as x approaches positive and negative infinity.

Explore the end behavior of the rational function f(x) = (-6x^5 + 8x^4 + 3x) / (6x^2 - 4x - 5) by analyzing its highest degree terms. Learn how the degrees of the numerator and denominator affect the behavior as x approaches infinity and negative infinity. Suitable for high school students studying algebra and calculus.

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