1. **State the problem:** Find the equation of a circle with center at $(-3,-5)$ and radius $8$ in the form $x^2 + Ax + y^2 + By = C$. 2. **Recall the standard form of a circle's equation:** $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h,k)$ is the center and $r$ is the radius. 3. **Substitute the given center and radius:** $$ (x - (-3))^2 + (y - (-5))^2 = 8^2 $$ which simplifies to $$ (x + 3)^2 + (y + 5)^2 = 64 $$ 4. **Expand the squares:** $$ (x + 3)^2 = x^2 + 2 \cdot 3 \cdot x + 3^2 = x^2 + 6x + 9 $$ $$ (y + 5)^2 = y^2 + 2 \cdot 5 \cdot y + 5^2 = y^2 + 10y + 25 $$ 5. **Combine and write the equation:** $$ x^2 + 6x + 9 + y^2 + 10y + 25 = 64 $$ 6. **Group like terms and simplify:** $$ x^2 + 6x + y^2 + 10y = 64 - 9 - 25 $$ $$ x^2 + 6x + y^2 + 10y = 30 $$ 7. **Final equation:** $$ x^2 + 6x + y^2 + 10y = 30 $$ This matches the form $x^2 + Ax + y^2 + By = C$ with $A=6$, $B=10$, and $C=30$. **Answer:** $$ x^2 + 6x + y^2 + 10y = 30 $$