1. **State the problem:** Simplify the expression $$-2x(4x + 6x^2 - 5x^3 + 12) - 5(x - 2) + (7 - 2x)(5 - 3x^2)$$. 2. **Distribute terms:** - Multiply $$-2x$$ by each term inside the first parentheses: $$-2x \cdot 4x = -8x^2$$ $$-2x \cdot 6x^2 = -12x^3$$ $$-2x \cdot (-5x^3) = +10x^4$$ $$-2x \cdot 12 = -24x$$ - Multiply $$-5$$ by each term inside the second parentheses: $$-5 \cdot x = -5x$$ $$-5 \cdot (-2) = +10$$ - Multiply the binomials $$(7 - 2x)(5 - 3x^2)$$ using FOIL: $$7 \cdot 5 = 35$$ $$7 \cdot (-3x^2) = -21x^2$$ $$-2x \cdot 5 = -10x$$ $$-2x \cdot (-3x^2) = +6x^3$$ 3. **Rewrite the expression with all terms:** $$-8x^2 - 12x^3 + 10x^4 - 24x - 5x + 10 + 35 - 21x^2 - 10x + 6x^3$$ 4. **Combine like terms:** - For $$x^4$$: $$10x^4$$ - For $$x^3$$: $$-12x^3 + 6x^3 = -6x^3$$ - For $$x^2$$: $$-8x^2 - 21x^2 = -29x^2$$ - For $$x$$: $$-24x - 5x - 10x = -39x$$ - Constants: $$10 + 35 = 45$$ 5. **Final simplified expression:** $$10x^4 - 6x^3 - 29x^2 - 39x + 45$$