1. **State the problem:** We need to find the value of $c$ in the equation of a circle given by $$x^2 + y^2 + ax + by + c = 0,$$ where the circle has center $(-5, 2)$ and radius $9$. 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. **Write the given circle's equation in standard form:** $$(x + 5)^2 + (y - 2)^2 = 9^2 = 81.$$ 4. **Expand the squared terms:** $$x^2 + 2 \cdot 5 \cdot x + 5^2 + y^2 - 2 \cdot 2 \cdot y + 2^2 = 81,$$ which simplifies to $$x^2 + 10x + 25 + y^2 - 4y + 4 = 81.$$ 5. **Combine like terms and rearrange to the general form:** $$x^2 + y^2 + 10x - 4y + (25 + 4 - 81) = 0,$$ $$x^2 + y^2 + 10x - 4y - 52 = 0.$$ 6. **Identify the constants:** Comparing with $$x^2 + y^2 + ax + by + c = 0,$$ we have $a = 10$, $b = -4$, and $c = -52$. **Final answer:** $$c = -52.$$