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The given problem is asking for the sum of the extreme values (극댓값 and 극솟값) of the function \( f(x) = -3x^4 - 4x^3 + 12x^2 - 15 \).

To solve this, we need to follow these steps:

### 1. **Find the derivative \( f'(x) \)**:
This will help us find the critical points (where extreme values occur).

\[
f(x) = -3x^4 - 4x^3 + 12x^2 - 15
\]
The derivative of \( f(x) \) is:
\[
f'(x) = -12x^3 - 12x^2 + 24x
\]

### 2. **Find critical points by setting \( f'(x) = 0 \)**:
\[
-12x^3 - 12x^2 + 24x = 0
\]

We can factor out \( -12x \) to solve this:
\[
-12x(x^2 + x - 2) = 0
\]
Now solve for \( x \):
\[
x(x^2 + x - 2) = 0
\]
This gives:
\[
x = 0 \quad \text{or} \quad x^2 + x - 2 = 0
\]

Solving \( x^2 + x - 2 = 0 \) using the quadratic formula:
\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 8}}{2} = \frac{-1 \pm \sqrt{9}}{2}
\]
\[
x = \frac{-1 \pm 3}{2}
\]
So, \( x = 1 \) or \( x = -2 \).

### 3. **Find the values of \( f(x) \) at the critical points**:
Now, substitute \( x = 0 \), \( x = 1 \), and \( x = -2 \) into the original function to find the extreme values.

- For \( x = 0 \):
\[
f(0) = -3(0)^4 - 4(0)^3 + 12(0)^2 - 15 = -15
\]

- For \( x = 1 \):
\[
f(1) = -3(1)^4 - 4(1)^3 + 12(1)^2 - 15 = -3 - 4 + 12 - 15 = -10
\]

- For \( x = -2 \):
\[
f(-2) = -3(-2)^4 - 4(-2)^3 + 12(-2)^2 - 15 = -3(16) - 4(-8) + 12(4) - 15 = -48 + 32 + 48 - 15 = 17
\]

### 4. **Sum of the extreme values**:
The extreme values are \( -15 \), \( -10 \), and \( 17 \). Their sum is:
\[
-15 + (-10) + 17 = -8
\]

### Final Answer:
The sum of the extreme values is \( \boxed{-8} \).

---

Would you like more details or have any questions about the process? Here are five related questions you might find helpful:

1. How do we interpret the critical points in terms of maxima and minima?
2. What is the significance of taking the second derivative for classifying extrema?
3. Can you explain the use of the quadratic formula in this context?
4. Why is factoring useful when solving derivative equations?
5. How do you verify whether a critical point is a local maximum or minimum?

**Tip**: When solving for extrema, always remember to substitute the critical points back into the original function, not its derivative.

함수 f(x) = -3x^4 - 4x^3 + 12x^2 - 15의 극댓값과 극솟값의 합을 구하여라.

In this example, learn how to calculate the sum of extreme values of the function f(x) = -3x^4 - 4x^3 + 12x^2 - 15. We use the first derivative to find critical points and apply the quadratic formula to solve for extrema.

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