1. **State the problem:** Find the x-coordinates where the graphs of $ f(x) = \cos^2(x) + 6\cos(x) $ and $ g(x) = -5 $ intersect for $ 0 \leq x < 2\pi $. 2. **Set the functions equal to find intersection points:** $$ \cos^2(x) + 6\cos(x) = -5 $$ 3. **Rewrite the equation:** $$ \cos^2(x) + 6\cos(x) + 5 = 0 $$ 4. **Use substitution:** Let $ y = \cos(x) $, then the equation becomes: $$ y^2 + 6y + 5 = 0 $$ 5. **Factor the quadratic:** $$ (y + 5)(y + 1) = 0 $$ 6. **Solve for $ y $:** $$ y = -5 \quad \text{or} \quad y = -1 $$ 7. **Check domain of cosine:** Since $ \cos(x) $ ranges from -1 to 1, $ y = -5 $ is invalid. 8. **Solve for $ x $ when $ \cos(x) = -1 $:** On $ [0, 2\pi) $, $ \cos(x) = -1 $ at: $$ x = \pi $$ 9. **Final answer:** The x-coordinate of the intersection point is: $$ \boxed{\pi} $$