The problem asks to convert the given center-radius form of a circle's equation into the general form. The provided equation is:

$(x+5)^2 + (y-3)^2 = 12$

To convert this to the general form $x^2 + y^2 + ax + by + c = 0$, we need to expand and simplify the given equation:

1. Expand $(x+5)^2$ and $(y-3)^2$: $(x+5)^2 = x^2 + 10x + 25$ $(y-3)^2 = y^2 - 6y + 9$

2. Add these expansions and set equal to 12: $x^2 + 10x + 25 + y^2 - 6y + 9 = 12$

3. Combine and simplify constants: $x^2 + 10x + y^2 - 6y + 25 + 9 - 12 = 0$ $x^2 + 10x + y^2 - 6y + 22 = 0$

Thus, the equation in general form is: $x^2 + y^2 + 10x - 6y + 22 = 0$

Among the given options:

• A. $x^2 + y^2 + 5x - 3y - 12 = 0$
• B. $x^2 + y^2 + 5x - 3y + 22 = 0$
• C. $x^2 + y^2 + 10x - 6y + 22 = 0$
• D. $x^2 + y^2 + 10x - 6y - 12 = 0$

Option C is the correct one.

Would you like more details or help with another question? Here are some other things you might want to explore:

1. How to derive the center and radius from the general form of a circle equation.
2. What changes if the center of the circle is shifted?
3. How to graph the equation of a circle given in general form.
4. How to find the intersection of this circle with a line.
5. How transformations like translations affect the equation of a circle.

Tip: When converting a circle's equation from center-radius form to general form, it's useful to fully expand and then simplify the terms step-by-step to avoid errors.