To find the general form of the equation of a circle with a given center and radius, follow these steps:

### Given:
- Center: \((-4, 6)\)
- Radius: \(5\)

### Step 1: Write the standard form of the circle's equation.
The standard form of a circle's equation is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

Substitute the given values:
\[
(x + 4)^2 + (y - 6)^2 = 5^2
\]
\[
(x + 4)^2 + (y - 6)^2 = 25
\]

### Step 2: Expand the equation.
Expand the squared terms:
\[
(x + 4)^2 = x^2 + 8x + 16
\]
\[
(y - 6)^2 = y^2 - 12y + 36
\]

Substitute these into the equation:
\[
x^2 + 8x + 16 + y^2 - 12y + 36 = 25
\]

### Step 3: Combine like terms.
Combine all the terms on one side to get the general form:
\[
x^2 + y^2 + 8x - 12y + 52 - 25 = 0
\]
\[
x^2 + y^2 + 8x - 12y + 27 = 0
\]

### General Form:
The general form of the equation is:
\[
x^2 + y^2 + 8x - 12y + 27 = 0
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you convert the general form of a circle's equation back to standard form?
2. What are the steps to find the radius and center from the general form of a circle?
3. Can a circle's equation be expressed in a different form, such as parametric?
4. How do you graph a circle using its general form equation?
5. What happens if the radius in the equation is negative or zero?

**Tip:** Always check your expanded terms carefully when converting between standard and general forms to avoid small algebraic errors.

Discover how to find the general form of a circle's equation with a given center at (-4, 6) and a radius of 5. This problem involves converting the standard form to the general form using key geometric concepts and equations. Suitable for high school students and advanced learners.

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