The problem provided asks to evaluate the limit:

$\lim_{{x \to -1}} (4x^3 + 5x^2 - 3x - 2)$

To solve this, we can directly substitute $x = -1$ into the polynomial since it's a continuous function.

Step-by-step evaluation:

$4(-1)^3 + 5(-1)^2 - 3(-1) - 2$

1. $(-1)^3 = -1$, so $4(-1)^3 = 4(-1) = -4$
2. $(-1)^2 = 1$, so $5(-1)^2 = 5(1) = 5$
3. $-3(-1) = 3$
4. The constant term is $-2$

Now, combine the terms:

$-4 + 5 + 3 - 2 = 2$

Therefore, the limit is:

$\lim_{{x \to -1}} (4x^3 + 5x^2 - 3x - 2) = 2$

The correct answer is 2.

Would you like further details or have any questions?

Here are some related questions for further practice:

1. How do you evaluate limits involving indeterminate forms?
2. What are the rules for continuous functions and limits?
3. Can you apply L'Hôpital's Rule to solve this limit?
4. What is the difference between limits at infinity and limits at a specific point?
5. How do higher-degree polynomials affect the behavior of limits?

Tip: For polynomial limits, direct substitution works when the function is continuous at the point of interest.