1. **State the problem:** We need to find all angles $x$ such that $0^\circ < x < 360^\circ$ and $\tan x = -2$. 2. **Recall the tangent function properties:** The tangent function has a period of $180^\circ$, meaning $\tan(x) = \tan(x + 180^\circ)$. Also, tangent is negative in the second and fourth quadrants. 3. **Find the reference angle:** First, find the angle whose tangent is $2$ (positive value) using the inverse tangent function: $$\theta = \tan^{-1}(2) \approx 63.435^\circ$$ 4. **Determine the angles where $\tan x = -2$:** Since tangent is negative in the second and fourth quadrants: - In the second quadrant: $x = 180^\circ - \theta = 180^\circ - 63.435^\circ = 116.565^\circ$ - In the fourth quadrant: $x = 360^\circ - \theta = 360^\circ - 63.435^\circ = 296.565^\circ$ 5. **Final answer:** The angles satisfying $\tan x = -2$ in $0^\circ < x < 360^\circ$ are: $$x = 116.565^\circ, 296.565^\circ$$