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To solve the equation \((x - 6)^2(x + 3) = 0\), we need to find the values of \(x\) that satisfy the equation.

### Step-by-Step Solution:

1. **Factor the Equation:**
   The equation is already factored: \((x - 6)^2(x + 3) = 0\).

2. **Apply the Zero Product Property:**
   According to the zero product property, if the product of several factors is zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero:

   \[
   (x - 6)^2 = 0 \quad \text{or} \quad (x + 3) = 0
   \]

3. **Solve Each Factor:**

   - For \((x - 6)^2 = 0\):
     \[
     x - 6 = 0
     \]
     \[
     x = 6
     \]

     Note: Since it's \((x - 6)^2\), the solution \(x = 6\) has a multiplicity of 2.

   - For \(x + 3 = 0\):
     \[
     x = -3
     \]

4. **Conclusion:**
   The solutions to the equation are \(x = 6\) and \(x = -3\).

   - \(x = 6\) (with multiplicity 2)
   - \(x = -3\) (with multiplicity 1)

These are the rational roots of the equation.

Would you like more details or have any questions?

### Related Questions:
1. How does the multiplicity of a root affect the graph of a polynomial function?
2. What is the Zero Product Property, and why is it important in solving polynomial equations?
3. How would you solve an equation that is not factored, such as \(x^2 + 3x - 18 = 0\)?
4. How do you determine the number of real roots for a given polynomial equation?
5. What are the general steps for solving quadratic equations by factoring?

### Tip:
When solving polynomial equations, always check if the equation can be factored easily. This can simplify the process significantly and help you find the roots more efficiently.

Discover how to solve the polynomial equation (x - 6)^2(x + 3) = 0 using the Zero Product Property. This detailed solution covers algebraic concepts such as factoring and solving quadratic equations. Ideal for students in Grades 9-10.

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