1. **State the problem:** Solve the trigonometric equation $$\cos x + \sqrt{2} = -\cos x$$ on the interval $$[0, 2\pi)$$. 2. **Rewrite the equation:** Move all terms involving $$\cos x$$ to one side: $$\cos x + \sqrt{2} = -\cos x \implies \cos x + \cos x = -\sqrt{2}$$ 3. **Combine like terms:** $$2\cos x = -\sqrt{2}$$ 4. **Isolate $$\cos x$$:** $$\cos x = \frac{-\sqrt{2}}{2}$$ 5. **Recall the cosine values:** $$\cos x = -\frac{\sqrt{2}}{2}$$ corresponds to angles where cosine is negative and equals $$-\frac{\sqrt{2}}{2}$$. 6. **Find reference angle:** The reference angle for $$\cos x = \frac{\sqrt{2}}{2}$$ is $$\frac{\pi}{4}$$. 7. **Determine solutions in $$[0, 2\pi)$$:** Cosine is negative in the second and third quadrants, so: $$x = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$ $$x = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$$ 8. **Write the final answer in increasing order:** $$x = \frac{3\pi}{4}, \frac{5\pi}{4}$$