1. **Problem 7:** Simplify the expression $ (5 - 2x^3 - 6x) - (2x^4 - 2x^2 + 5x^3 - 5) - (6x^3 + x^2 + 1 + 4x) $. 2. Write the expression without parentheses, changing signs for the terms in the subtracted groups: $$ 5 - 2x^3 - 6x - 2x^4 + 2x^2 - 5x^3 + 5 - 6x^3 - x^2 - 1 - 4x $$ 3. Combine like terms by grouping powers of $x$: - Constant terms: $5 + 5 - 1 = 9$ - $x^4$ term: $-2x^4$ - $x^3$ terms: $-2x^3 - 5x^3 - 6x^3 = -13x^3$ - $x^2$ terms: $2x^2 - x^2 = x^2$ - $x$ terms: $-6x - 4x = -10x$ 4. Final simplified expression: $$ -2x^4 - 13x^3 + x^2 - 10x + 9 $$ --- 5. **Problem 8:** Find the product $2m^4(8m - 5)$. 6. Use distributive property: $$ 2m^4 \times 8m - 2m^4 \times 5 = 16m^{4+1} - 10m^4 = 16m^5 - 10m^4 $$ --- 7. **Problem 9:** Find the product $8(3m - 7)$. 8. Distribute 8: $$ 8 \times 3m - 8 \times 7 = 24m - 56 $$ --- 9. **Problem 10:** Find the product $3(6a + 6)$. 10. Distribute 3: $$ 3 \times 6a + 3 \times 6 = 18a + 18 $$ --- 11. **Problem 11:** Find the product $8(4r^2 - 3r - 4)$. 12. Distribute 8: $$ 8 \times 4r^2 - 8 \times 3r - 8 \times 4 = 32r^2 - 24r - 32 $$ --- 13. **Problem 12:** Find the product $3(7n^2 - 7n - 6)$. 14. Distribute 3: $$ 3 \times 7n^2 - 3 \times 7n - 3 \times 6 = 21n^2 - 21n - 18 $$ --- 15. **Problem 13:** Find the product $(5r - 2)(7r - 6)$. 16. Use FOIL method: $$ 5r \times 7r + 5r \times (-6) + (-2) \times 7r + (-2) \times (-6) = 35r^2 - 30r - 14r + 12 $$ 17. Combine like terms: $$ 35r^2 - 44r + 12 $$ --- 18. **Problem 14:** Find the product $(7b - 6)(5b + 3)$. 19. Use FOIL: $$ 7b \times 5b + 7b \times 3 - 6 \times 5b - 6 \times 3 = 35b^2 + 21b - 30b - 18 $$ 20. Combine like terms: $$ 35b^2 - 9b - 18 $$ --- 21. **Problem 15:** Find the product $(6n + 5)(4n - 4)$. 22. Use FOIL: $$ 6n \times 4n + 6n \times (-4) + 5 \times 4n + 5 \times (-4) = 24n^2 - 24n + 20n - 20 $$ 23. Combine like terms: $$ 24n^2 - 4n - 20 $$