1. **State the problem:** Convert the Cartesian point $(-1, -2)$ to polar coordinates $(r, \theta)$, where $r = \sqrt{5}$ and $0 \leq \theta < 2\pi$. 2. **Formula used:** Polar coordinates are given by: $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\left(\frac{y}{x}\right)$$ Important: Adjust $\theta$ based on the quadrant of the point. 3. **Calculate $r$ to verify:** $$r = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$$ This matches the given $r$. 4. **Calculate $\theta$:** $$\theta = \arctan\left(\frac{-2}{-1}\right) = \arctan(2)$$ 5. **Determine the quadrant:** The point $(-1, -2)$ is in the third quadrant (both $x$ and $y$ negative). 6. **Adjust $\theta$ for third quadrant:** Since $\arctan(2)$ gives an angle in the first quadrant, add $\pi$ to get the correct angle: $$\theta = \arctan(2) + \pi$$ 7. **Final polar coordinates:** $$\boxed{\left(\sqrt{5}, \arctan(2) + \pi\right)}$$ This means the radius is $\sqrt{5}$ and the angle $\theta$ is $\arctan(2) + \pi$, which lies between $\pi$ and $2\pi$ as required.