1. **Stating the problem:** Solve the equation $$(x - 1)^2 = -4$$ for $x$. 2. **Understanding the equation:** The left side is a square of a real number $(x-1)$, which means it is always non-negative (i.e., $\geq 0$). 3. **Important rule:** A square of a real number cannot be negative. Since the right side is $-4$, which is negative, there is no real number $x$ that satisfies this equation. 4. **Conclusion:** The equation has no real solutions because $$(x - 1)^2 \geq 0$$ for all real $x$, but the equation requires it to be $-4$. 5. **If considering complex numbers:** We can solve by taking the square root of both sides: $$x - 1 = \pm \sqrt{-4} = \pm 2i$$ Then, $$x = 1 \pm 2i$$ where $i$ is the imaginary unit. **Final answer:** - No real solutions. - Complex solutions: $$x = 1 + 2i$$ and $$x = 1 - 2i$$.