The given equation is:

\[
(x + 2)^2 + (y - 1)^2 = 9
\]

This is the equation of a circle in standard form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

### Analyzing the equation:

1. **Center**: The equation can be rewritten to match the standard form as:
   \[
   (x - (-2))^2 + (y - 1)^2 = 3^2
   \]
   Thus, the center of the circle is at \((-2, 1)\).

2. **Radius**: The radius \(r\) is the square root of 9, which gives:
   \[
   r = \sqrt{9} = 3
   \]

### Summary:
- **Center**: \((-2, 1)\)
- **Radius**: \(3\)

Would you like more details or have any further questions?

Here are some related questions:

1. How do you graph this circle?
2. What is the area of the circle?
3. How do you find the circumference of the circle?
4. How would the equation change if the circle were moved to the origin?
5. What if the radius were doubled? How would the equation look?
6. How do you derive the equation of a circle from its geometric definition?
7. What other geometric shapes can be expressed in a similar equation form?
8. How does the equation change if the circle is scaled by a factor of 2?

**Tip:** When dealing with the equation of a circle, always identify the center and the radius first to understand the geometric properties easily.

Learn how to solve the circle equation (x + 2)^2 + (y - 1)^2 = 9 with a detailed explanation of finding the center and radius. This problem involves analyzing the standard form of a circle equation and its geometric interpretation. Suitable for high school students.

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