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We are given two functions:

- \( f(x) = 3|x + 1| - 4 \)
- \( g(x) = -2x^2 - 5 \)

We are asked to find \( f(g(x)) \), which means we will substitute \( g(x) \) into the function \( f(x) \).

### Step-by-step solution:

1. \( g(x) = -2x^2 - 5 \), so we substitute \( g(x) \) into \( f(x) \):
   \[
   f(g(x)) = 3 \left| g(x) + 1 \right| - 4
   \]
   Substituting \( g(x) = -2x^2 - 5 \):
   \[
   f(g(x)) = 3 \left| (-2x^2 - 5) + 1 \right| - 4
   \]

2. Simplify the expression inside the absolute value:
   \[
   (-2x^2 - 5) + 1 = -2x^2 - 4
   \]
   So, we have:
   \[
   f(g(x)) = 3 \left| -2x^2 - 4 \right| - 4
   \]

3. Since the absolute value of \( -2x^2 - 4 \) is \( 2x^2 + 4 \) (because the absolute value removes the negative sign), we have:
   \[
   f(g(x)) = 3(2x^2 + 4) - 4
   \]

4. Now, distribute the 3:
   \[
   f(g(x)) = 6x^2 + 12 - 4
   \]

5. Simplify the expression:
   \[
   f(g(x)) = 6x^2 + 8
   \]

Thus, \( f(g(x)) = 6x^2 + 8 \).

---

Would you like any further clarifications?

Here are five related questions to further your understanding:
1. What happens if you change the sign inside the absolute value?
2. How would the answer change if \( g(x) \) were linear instead of quadratic?
3. Can you apply this process to a different pair of functions, say \( f(x) = x^2 \) and \( g(x) = x + 1 \)?
4. What is the significance of the absolute value in the function \( f(x) \)?
5. How would you differentiate \( f(g(x)) \) with respect to \( x \)?

### Tip:
Always handle absolute value expressions carefully by considering both positive and negative cases of the content inside the absolute value.

If f(x) = 3|x + 1| - 4 and g(x) = -2x^2 - 5, what is f(g(x))?

Learn how to solve a function composition problem where f(x) = 3|x + 1| - 4 and g(x) = -2x^2 - 5. This step-by-step solution explains the process of substituting one function into another, focusing on absolute value and quadratic expressions. Suitable for students in Grades 10-12.

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