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The problem asks for a fourth-degree polynomial \( g(x) \) with specific characteristics:

- A double zero at \( x = -2 \), meaning \( g(x) \) has the factor \( (x + 2)^2 \).
- Zeros at \( x = -5 \) and \( x = 6 \), meaning \( g(x) \) also has the factors \( (x + 5) \) and \( (x - 6) \).
- Additionally, \( g(0) = -12 \), which allows us to find the leading coefficient.

### Step-by-step Solution:

1. **Construct the general form:**
   Since the polynomial has a double zero at \( x = -2 \), a zero at \( x = -5 \), and a zero at \( x = 6 \), the general form of the polynomial is:
   \[
   g(x) = a(x + 2)^2(x + 5)(x - 6)
   \]
   where \( a \) is a constant coefficient that we will determine.

2. **Use the condition \( g(0) = -12 \):**
   Substituting \( x = 0 \) into the polynomial:
   \[
   g(0) = a(0 + 2)^2(0 + 5)(0 - 6)
   \]
   Simplifying this:
   \[
   g(0) = a(4)(5)(-6) = -120a
   \]
   Since \( g(0) = -12 \), we have:
   \[
   -120a = -12
   \]
   Solving for \( a \):
   \[
   a = \frac{-12}{-120} = \frac{1}{10}
   \]

3. **Final polynomial:**
   Substituting \( a = \frac{1}{10} \) back into the general form of the polynomial:
   \[
   g(x) = \frac{1}{10}(x + 2)^2(x + 5)(x - 6)
   \]

Thus, a possible formula for \( g(x) \) is:
\[
g(x) = \frac{1}{10}(x + 2)^2(x + 5)(x - 6)
\]

Let me know if you need further details or a deeper explanation!

### Related Questions:
1. How do we determine the degree of a polynomial from its factors?
2. What is the significance of having a "double zero" at a particular point?
3. Can you explain how changing the constant \( a \) affects the graph of the polynomial?
4. How would the polynomial change if one of the zeros was a triple zero?
5. What is the general strategy to find the equation of a polynomial given its roots and a specific function value?

### Tip:
Always check the highest degree and number of zeros to ensure the polynomial matches the required degree!

Find a possible formula for a fourth-degree polynomial g that has a double zero at -2, g(-5) = 0, g(6) = 0, and g(0) = -12.

Solve a fourth-degree polynomial equation with a double zero at x = -2, and zeros at x = -5 and x = 6. This problem explores polynomial roots, factoring, and solving for the leading coefficient using g(0) = -12.

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