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We are given the following limits:

\[
\lim_{{x \to a}} f(x) = 3 \quad \text{and} \quad \lim_{{x \to a}} g(x) = -4,
\]

and we are asked to evaluate the limit:

\[
\lim_{{x \to a}} \frac{-2[g(x)]^2}{f(x) + 5}.
\]

### Step-by-step Solution:

1. **Substitute the limit values of \( f(x) \) and \( g(x) \):**

\[
f(x) \to 3 \quad \text{and} \quad g(x) \to -4 \quad \text{as} \quad x \to a.
\]

2. **Evaluate the expression using these limit values:**

\[
\lim_{{x \to a}} \frac{-2[g(x)]^2}{f(x) + 5} = \frac{-2[-4]^2}{3 + 5}.
\]

3. **Simplify the expression:**

- First, calculate \([-4]^2\):
\[
[-4]^2 = 16.
\]

- Now, substitute back:
\[
\frac{-2(16)}{3 + 5} = \frac{-32}{8}.
\]

4. **Final simplification:**

\[
\frac{-32}{8} = -4.
\]

Thus, the value of the limit is \(-4\).

### Answer:
The correct choice is **A. \(-4\)**.

---

Would you like more details or have any questions? Here are some related questions for further understanding:

1. What is the general procedure for evaluating limits involving fractions?
2. How do squared terms affect the outcome of limit problems?
3. What happens if one of the functions \( f(x) \) or \( g(x) \) approaches 0?
4. How would the limit change if the denominator approached 0?
5. Can limits of composite functions always be computed by substitution?

**Tip:** When solving limit problems, always check if direct substitution simplifies the expression. If not, consider algebraic simplification or limit laws.

If lim f(x) = 3 and lim g(x) = -4, find lim [ -2(g(x))^2 / (f(x) + 5) ]

This problem demonstrates how to evaluate the limit of a complex expression involving functions f(x) and g(x) using substitution. Given lim f(x) = 3 and lim g(x) = -4, the final limit is evaluated to -4.

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