1. **State the problem:** Solve the equation $\tan 4x = -1$ for $x$. 2. **Recall the formula and properties:** The tangent function satisfies $\tan \theta = -1$ at angles where $\theta = -\frac{\pi}{4} + k\pi$ for any integer $k$. 3. **Set the argument equal to these angles:** $$4x = -\frac{\pi}{4} + k\pi$$ 4. **Solve for $x$ by dividing both sides by 4:** $$x = \frac{-\frac{\pi}{4} + k\pi}{4} = \frac{-\pi/4}{4} + \frac{k\pi}{4} = -\frac{\pi}{16} + \frac{k\pi}{4}$$ 5. **Intermediate step showing cancellation:** $$x = \frac{\cancel{4}x}{\cancel{4}} = \frac{-\frac{\pi}{4} + k\pi}{4}$$ 6. **Final solution:** $$\boxed{x = -\frac{\pi}{16} + \frac{k\pi}{4}, \quad k \in \mathbb{Z}}$$ This means $x$ takes all values of the form $-\frac{\pi}{16} + \frac{k\pi}{4}$ where $k$ is any integer.