1. **State the problem:** Solve the quadratic equation $$-2x^2 + 4x - 4 = 0$$ by completing the square. 2. **Rewrite the equation:** First, divide the entire equation by -2 to make the coefficient of $x^2$ equal to 1. $$\cancel{-2}x^2 + \cancel{4}x - \cancel{4} = 0 \implies x^2 - 2x + 2 = 0$$ 3. **Complete the square:** Take the coefficient of $x$, which is $-2$, divide it by 2 to get $-1$, then square it to get $1$. 4. Add and subtract this square inside the equation to keep it balanced: $$x^2 - 2x + 1 - 1 + 2 = 0$$ 5. Group the perfect square trinomial and simplify: $$ (x - 1)^2 + 1 = 0$$ 6. **Solve for $x$:** Subtract 1 from both sides: $$ (x - 1)^2 = -1$$ 7. Since the right side is negative, the solutions are complex numbers: $$ x - 1 = \pm \sqrt{-1} = \pm i$$ 8. Finally, solve for $x$: $$ x = 1 \pm i$$ **Answer:** The solutions are $$x = 1 + i$$ and $$x = 1 - i$$.