1. **State the problem:** We are given two points in polar coordinates, $P_1 = (-3, \frac{5\pi}{6})$ and $P_2 = (4, \frac{4\pi}{3})$, and we need to find the area of the triangle formed by these points and the pole (origin). 2. **Formula used:** The area of a triangle formed by two points in polar coordinates and the origin is given by: $$\text{Area} = \frac{1}{2} r_1 r_2 \sin(\theta_2 - \theta_1)$$ where $r_1, r_2$ are the radii and $\theta_1, \theta_2$ are the angles of the points. 3. **Important note:** Negative radius means the point is in the opposite direction of the angle. We convert $P_1 = (-3, \frac{5\pi}{6})$ to an equivalent positive radius and angle by adding $\pi$ to the angle and taking positive radius: $$r_1 = 3, \quad \theta_1 = \frac{5\pi}{6} + \pi = \frac{11\pi}{6}$$ 4. **Calculate the difference in angles:** $$\theta_2 - \theta_1 = \frac{4\pi}{3} - \frac{11\pi}{6} = \frac{8\pi}{6} - \frac{11\pi}{6} = -\frac{3\pi}{6} = -\frac{\pi}{2}$$ 5. **Calculate the sine:** $$\sin\left(-\frac{\pi}{2}\right) = -1$$ 6. **Calculate the area:** $$\text{Area} = \frac{1}{2} \times 3 \times 4 \times |-1| = \frac{1}{2} \times 12 = 6$$ 7. **Final answer:** The area of the triangle formed by $P_1$, $P_2$, and the pole is **6** square units.