1. **State the problem:** Simplify the algebraic expression $$-2y^3 + 0.5x^2y - 12x^2 - 4zy + 3x^3 - 4x^2y - 3x^2 - 4zy + 5y^3 - 3x^2 + 3x^3$$. 2. **Group like terms:** Group terms with the same variables and powers. - For $y^3$: $$-2y^3 + 5y^3$$ - For $x^2y$: $$0.5x^2y - 4x^2y$$ - For $x^2$: $$-12x^2 - 3x^2 - 3x^2$$ - For $zy$: $$-4zy - 4zy$$ - For $x^3$: $$3x^3 + 3x^3$$ 3. **Simplify each group:** - $y^3$ terms: $$-2y^3 + 5y^3 = ( -2 + 5 ) y^3 = 3y^3$$ - $x^2y$ terms: $$0.5x^2y - 4x^2y = (0.5 - 4) x^2y = -3.5x^2y$$ - $x^2$ terms: $$-12x^2 - 3x^2 - 3x^2 = (-12 - 3 - 3) x^2 = -18x^2$$ - $zy$ terms: $$-4zy - 4zy = (-4 - 4) zy = -8zy$$ - $x^3$ terms: $$3x^3 + 3x^3 = (3 + 3) x^3 = 6x^3$$ 4. **Write the simplified expression:** $$3y^3 - 3.5x^2y - 18x^2 - 8zy + 6x^3$$ This is the simplified form of the original expression. **Final answer:** $$3y^3 - 3.5x^2y - 18x^2 - 8zy + 6x^3$$