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Let's break down and solve the problem shown in the image.

### Problem:
Find a polynomial of degree 3 with real coefficients and zeros of \(-3\), \(-1\), and \(4\), for which \(f(-2) = 24\).

### Step-by-Step Solution:

1. **Form the Polynomial**:
   Since the zeros of the polynomial are \(-3\), \(-1\), and \(4\), the factors of the polynomial are:
   \[
   (x + 3), (x + 1), (x - 4)
   \]
   The general form of the polynomial is:
   \[
   f(x) = a(x + 3)(x + 1)(x - 4)
   \]
   where \(a\) is a constant to be determined.

2. **Use the Condition \(f(-2) = 24\)**:
   To find \(a\), we substitute \(x = -2\) into the polynomial and set it equal to 24:
   \[
   f(-2) = a(-2 + 3)(-2 + 1)(-2 - 4) = 24
   \]
   Simplifying each factor:
   \[
   f(-2) = a(1)(-1)(-6) = 24
   \]
   \[
   -6a = 24
   \]
   Solving for \(a\):
   \[
   a = \frac{24}{-6} = -4
   \]

3. **Write the Final Polynomial**:
   Now that we know \(a = -4\), the polynomial is:
   \[
   f(x) = -4(x + 3)(x + 1)(x - 4)
   \]

4. **Expand the Polynomial (Optional)**:
   If you want to simplify the polynomial further, you can expand it:
   \[
   f(x) = -4[(x + 3)(x + 1)(x - 4)]
   \]
   First, expand \((x + 3)(x + 1)\):
   \[
   (x + 3)(x + 1) = x^2 + 4x + 3
   \]
   Now multiply this by \((x - 4)\):
   \[
   (x^2 + 4x + 3)(x - 4) = x^3 - 4x^2 + 4x^2 - 16x + 3x - 12
   \]
   Simplify:
   \[
   = x^3 - 13x - 12
   \]
   Finally, multiply by \(-4\):
   \[
   f(x) = -4(x^3 - 13x - 12) = -4x^3 + 52x + 48
   \]
   So the polynomial is:
   \[
   f(x) = -4x^3 + 52x + 48
   \]

This is the polynomial you're looking for. Do you need further clarification on any step?

---

### Related Questions:
1. How do you find the roots of a cubic polynomial?
2. What is the importance of the constant \(a\) in defining a polynomial?
3. Can the polynomial have complex roots if the coefficients are real?
4. What methods can be used to factor a cubic polynomial directly?
5. How does the transformation \(f(-2) = 24\) affect the graph of the polynomial?

### Tip:
When given roots, always form the polynomial using the corresponding factors and then adjust for any scaling conditions using additional information provided (like \(f(x)\) values).

Find a polynomial of degree 3 with real coefficients and zeros of -3, -1, and 4, for which f(-2) = 24.

Learn how to find a degree 3 polynomial with given zeros of -3, -1, and 4, and a condition that f(-2) = 24. This step-by-step solution covers the use of polynomial roots, substitution, and expansion.

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