1. **Stating the problem:** Find the equation of a circle with center at $(5, -2)$ and radius $4$ cm. 2. **Formula for the equation of a circle:** The general equation of a circle with center $(h, k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. **Substitute the given values:** Here, $h = 5$, $k = -2$, and $r = 4$. 4. **Write the equation:** $$ (x - 5)^2 + (y - (-2))^2 = 4^2 $$ 5. **Simplify:** $$ (x - 5)^2 + (y + 2)^2 = 16 $$ 6. **Final answer:** The equation of the circle is: $$ (x - 5)^2 + (y + 2)^2 = 16 $$ This equation represents all points $(x, y)$ that are exactly 4 units away from the center $(5, -2)$.