1. **State the problem:** Multiply the expressions $(-2x)$ and $(3x^4)$. 2. **Write the multiplication:** $$(-2x)(3x^4)$$ 3. **Multiply the coefficients:** $$-2 \times 3 = -6$$ 4. **Multiply the variables:** $$x \times x^4 = x^{1+4} = x^5$$ 5. **Combine the results:** $$-6x^5$$ 6. **Final answer:** The product of $(-2x)$ and $(3x^4)$ is $$\boxed{-6x^5}$$