1. **State the problem:** Solve the inequality $$x^3 - 4x^2 - x + 4 \geq 0$$. 2. **Find the roots of the cubic polynomial:** To solve the inequality, first find the roots of the equation $$x^3 - 4x^2 - x + 4 = 0$$. 3. **Try rational root theorem candidates:** Possible rational roots are factors of 4: $$\pm1, \pm2, \pm4$$. 4. **Test roots:** - For $$x=1$$: $$1 - 4 - 1 + 4 = 0$$, so $$x=1$$ is a root. 5. **Factor out $$x-1$$:** Use polynomial division or synthetic division: $$x^3 - 4x^2 - x + 4 = (x-1)(x^2 - 3x - 4)$$. 6. **Factor the quadratic:** $$x^2 - 3x - 4 = (x - 4)(x + 1)$$. 7. **Roots are:** $$x=1, x=4, x=-1$$. 8. **Determine sign intervals:** The polynomial factors as $$ (x-1)(x-4)(x+1) \geq 0 $$. 9. **Test intervals between roots:** - For $$x < -1$$, pick $$x=-2$$: $$(-)(-)(-) = -$$ (negative) - For $$-1 < x < 1$$, pick $$x=0$$: $$(-)(-)(+) = +$$ (positive) - For $$1 < x < 4$$, pick $$x=2$$: $$(+)(-)(+) = -$$ (negative) - For $$x > 4$$, pick $$x=5$$: $$(+)(+)(+) = +$$ (positive) 10. **Include roots where polynomial equals zero:** At $$x=-1, 1, 4$$, polynomial equals zero. 11. **Solution:** The inequality $$x^3 - 4x^2 - x + 4 \geq 0$$ holds for $$[-1,1] \cup [4, \infty)$$. **Final answer:** $$\boxed{[-1,1] \cup [4, \infty)}$$