1. **State the problem:** Simplify the expression $$(1 - 4x^3) - (5 - 3x^3)$$. 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to each term inside the second parentheses. $$1 - 4x^3 - 5 + 3x^3$$ 3. **Group like terms:** Combine the constant terms and the terms with $x^3$. Constants: $$1 - 5 = \cancel{1} - \cancel{5} = -4$$ $x^3$ terms: $$-4x^3 + 3x^3 = \cancel{-4x^3} + \cancel{3x^3} = -x^3$$ 4. **Write the simplified expression:** $$-4 - x^3$$ 5. **Final answer:** $$\boxed{-4 - x^3}$$