The given equation is:

$(x - 11)^2 + (y + 13)^2 = 25$

This represents the equation of a circle in standard form, where the center is $(h, k) = (11, -13)$, and the radius is $\sqrt{25} = 5$.

To convert this into the general form, we will expand both binomials and rearrange terms.

Step-by-step solution:

1. Expand the binomials: $(x - 11)^2 = x^2 - 22x + 121$ $(y + 13)^2 = y^2 + 26y + 169$

2. Substituting these expansions back into the equation: $x^2 - 22x + 121 + y^2 + 26y + 169 = 25$

3. Combine like terms: $x^2 - 22x + y^2 + 26y + 121 + 169 = 25$ $x^2 - 22x + y^2 + 26y + 290 = 25$

4. Move everything to one side to form the general equation: $x^2 - 22x + y^2 + 26y + 290 - 25 = 0$ $x^2 - 22x + y^2 + 26y + 265 = 0$

This matches the first option in the image.

Thus, the correct answer is:

$x^2 - 22x + y^2 + 26y + 265 = 0$

Let me know if you need further clarification or details!

Related Questions:

1. How do you derive the standard form of a circle from the general form?
2. What is the geometric interpretation of the general form of a circle?
3. How can you determine the radius of a circle given its general form?
4. What is the process to find the center of a circle from its equation?
5. How do we convert a quadratic equation in two variables into standard form?

Tip: Always check if the constants in your equation match both sides when simplifying!