1. **State the problem:** Solve the equation $$\tan^3(x) + 3\tan(x) = 0$$ for $x$. 2. **Rewrite the equation:** Factor out $\tan(x)$: $$\tan(x)(\tan^2(x) + 3) = 0$$ 3. **Set each factor equal to zero:** - First factor: $$\tan(x) = 0$$ - Second factor: $$\tan^2(x) + 3 = 0$$ 4. **Solve the first factor:** $$\tan(x) = 0$$ The tangent function is zero at integer multiples of $\pi$: $$x = k\pi, \quad k \in \mathbb{Z}$$ 5. **Solve the second factor:** $$\tan^2(x) + 3 = 0 \implies \tan^2(x) = -3$$ Since $\tan^2(x)$ is always non-negative, it cannot equal $-3$. Therefore, no real solutions come from this factor. 6. **Final solution:** $$x = k\pi, \quad k \in \mathbb{Z}$$ This means the solutions are all integer multiples of $\pi$ where the tangent function is zero.