1. **State the problem:** Find the distance $d(P_1, P_2)$ between the points $P_1 = (2, 5)$ and $P_2 = (-2, -4)$. 2. **Formula used:** The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Substitute the values:** $$d = \sqrt{(-2 - 2)^2 + (-4 - 5)^2}$$ 4. **Simplify inside the square root:** $$d = \sqrt{(-4)^2 + (-9)^2}$$ 5. **Calculate the squares:** $$d = \sqrt{16 + 81}$$ 6. **Add the values:** $$d = \sqrt{97}$$ 7. **Conclusion:** The distance between the points is $\sqrt{97}$. **Final answer:** $\sqrt{97}$