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The problem asks for the following:

1. **Graph the circle.**
2. **Determine the general equation of the circle.**
3. **Find the center and radius of the circle.**

The given equation of the circle is:

\[
(x + 3)^2 + (y - 4)^2 = 64
\]

### 1. **Center and Radius of the Circle**
The equation \((x + 3)^2 + (y - 4)^2 = 64\) is in the standard form of a circle's equation:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where:
- \((h, k)\) is the center of the circle,
- \(r\) is the radius of the circle.

From the given equation, we can identify:
- \(h = -3\) (since \(x + 3\) is the same as \(x - (-3)\)),
- \(k = 4\),
- \(r^2 = 64\), so \(r = \sqrt{64} = 8\).

Thus, the **center** of the circle is \((-3, 4)\) and the **radius** is \(8\) units.

### 2. **General Equation of the Circle**
To find the general form of the equation, expand the given equation:

\[
(x + 3)^2 + (y - 4)^2 = 64
\]

Expanding both squared terms:

\[
(x^2 + 6x + 9) + (y^2 - 8y + 16) = 64
\]

Combine all terms:

\[
x^2 + y^2 + 6x - 8y + 25 = 64
\]

Finally, subtract \(64\) from both sides to get the equation in general form:

\[
x^2 + y^2 + 6x - 8y - 39 = 0
\]

### 3. **Graphing the Circle**
The circle can be graphed by plotting the center at \((-3, 4)\) and drawing a circle with a radius of \(8\) units.

Would you like a more detailed explanation or further questions on this topic? Here are some related questions:

1. What is the general form of a conic section, and how does it apply to circles?
2. How do you find the equation of a circle given a different form, like the diameter endpoints?
3. Can the center-radius form of a circle equation be derived from its general form?
4. How does the process change if the equation was not in standard form initially?
5. What is the geometric significance of the coefficients in the general form of a circle's equation?
6. How do transformations affect the equation of a circle?
7. What are the steps to graph a circle given its equation?
8. Can you determine the equation of a circle given three non-collinear points?

**Tip:** When identifying the center and radius of a circle from its equation, always check if the equation is in the standard form \((x - h)^2 + (y - k)^2 = r^2\) first, and then adjust the center coordinates accordingly.

Explore how to graph the circle with equation (x + 3)^2 + (y - 4)^2 = 64, find its center and radius, and derive its general equation. This detailed math solution covers circle geometry, coordinate geometry, and the standard form of a circle's equation. Suitable for high school students.

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