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To solve the given limit problem:

\[
\lim_{{x \to -\infty}} \frac{-6x - 4}{8x^3 - 6x^2 + x + 4}
\]

### Step-by-step process:

1. **Examine the degrees of the polynomial**:
   - In the numerator: \(-6x - 4\) has a degree of 1.
   - In the denominator: \(8x^3 - 6x^2 + x + 4\) has a degree of 3.

2. **Leading terms for large \(x\)**:
   - The leading term in the numerator is \(-6x\).
   - The leading term in the denominator is \(8x^3\).

3. **Simplify by dividing numerator and denominator by \(x^3\)** (the highest power of \(x\) in the denominator):
   \[
   \frac{-6x - 4}{8x^3 - 6x^2 + x + 4} = \frac{\frac{-6x - 4}{x^3}}{\frac{8x^3 - 6x^2 + x + 4}{x^3}}
   \]

4. **Simplify each term**:
   - Numerator: \(\frac{-6x - 4}{x^3} = \frac{-6}{x^2} - \frac{4}{x^3}\).
   - Denominator: \(\frac{8x^3}{x^3} - \frac{6x^2}{x^3} + \frac{x}{x^3} + \frac{4}{x^3} = 8 - \frac{6}{x} + \frac{1}{x^2} + \frac{4}{x^3}\).

5. **Take the limit as \(x \to -\infty\)**:
   - The terms involving \(x^{-1}\), \(x^{-2}\), and \(x^{-3}\) in both the numerator and denominator will approach 0 as \(x \to -\infty\).
   - Thus, the numerator approaches 0.
   - The denominator approaches \(8\).

6. **Conclusion**:
   The limit is:
   \[
   \lim_{{x \to -\infty}} \frac{0}{8} = 0
   \]

Therefore, the correct answer is **0**.

Would you like more details on any step, or do you have any questions?

### Related Questions:
1. How do you determine which term dominates in the limit at infinity?
2. What would happen if the degrees of the numerator and denominator were the same?
3. How do you handle limits approaching infinity versus negative infinity?
4. What happens when the numerator has a higher degree than the denominator?
5. Can we use L'Hopital's rule to solve this limit?

**Tip**: When solving limits involving polynomials, always look for the highest degree terms, as they dominate the behavior at large values of \(x\).

Compute the limit lim(x → -∞) ((-6x - 4) / (8x^3 - 6x^2 + x + 4))

Learn how to compute the limit of a rational function (-6x - 4) / (8x^3 - 6x^2 + x + 4) as x approaches negative infinity. This problem involves polynomial simplification and leading term analysis.

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