1. **State the problem:** Simplify the expression $$3y(5 - 8y^2)(4x + 8y) - (3 - 5y)(20 - 7) + 5y^2$$. 2. **Apply the distributive property and simplify constants:** Calculate $$20 - 7 = 13$$. So the expression becomes: $$3y(5 - 8y^2)(4x + 8y) - (3 - 5y)(13) + 5y^2$$. 3. **Expand the first product:** First expand $$(5 - 8y^2)(4x + 8y)$$: $$5 \cdot 4x = 20x$$ $$5 \cdot 8y = 40y$$ $$-8y^2 \cdot 4x = -32xy^2$$ $$-8y^2 \cdot 8y = -64y^3$$ So, $$(5 - 8y^2)(4x + 8y) = 20x + 40y - 32xy^2 - 64y^3$$. 4. **Multiply by $3y$:** $$3y(20x + 40y - 32xy^2 - 64y^3) = 3y \cdot 20x + 3y \cdot 40y - 3y \cdot 32xy^2 - 3y \cdot 64y^3$$ $$= 60xy + 120y^2 - 96xy^3 - 192y^4$$. 5. **Expand the second product:** $$(3 - 5y)(13) = 3 \cdot 13 - 5y \cdot 13 = 39 - 65y$$. 6. **Rewrite the entire expression:** $$60xy + 120y^2 - 96xy^3 - 192y^4 - (39 - 65y) + 5y^2$$. 7. **Distribute the minus sign:** $$60xy + 120y^2 - 96xy^3 - 192y^4 - 39 + 65y + 5y^2$$. 8. **Combine like terms:** Combine $$120y^2 + 5y^2 = 125y^2$$. Final simplified expression: $$60xy + 125y^2 - 96xy^3 - 192y^4 + 65y - 39$$.