1. **Problem 2.224 a)** Simplify $ \frac{3a^2 + 2}{5a + 5} + \frac{1 - a}{10} $. 2. Factor the denominator in the first fraction: $5a + 5 = 5(a + 1)$. 3. Find the common denominator for both fractions: $10(a + 1)$. 4. Rewrite each fraction with the common denominator: $$\frac{3a^2 + 2}{5(a + 1)} = \frac{2(3a^2 + 2)}{10(a + 1)}$$ $$\frac{1 - a}{10} = \frac{(1 - a)(a + 1)}{10(a + 1)}$$ 5. Add the numerators: $$2(3a^2 + 2) + (1 - a)(a + 1) = 6a^2 + 4 + (1 - a)(a + 1)$$ 6. Expand $(1 - a)(a + 1) = 1 \cdot a + 1 \cdot 1 - a \cdot a - a \cdot 1 = a + 1 - a^2 - a = 1 - a^2$$ 7. Sum the numerator: $$6a^2 + 4 + 1 - a^2 = 5a^2 + 5$$ 8. Factor numerator: $$5a^2 + 5 = 5(a^2 + 1)$$ 9. Final expression: $$\frac{5(a^2 + 1)}{10(a + 1)} = \frac{\cancel{5}(a^2 + 1)}{\cancel{10}2(a + 1)} = \frac{a^2 + 1}{2(a + 1)}$$ --- 10. **Problem 2.224 b)** Simplify \( \frac{1 + 2b}{8b + 4} - \frac{2b + 9}{8b} $. 11. Factor denominator $8b + 4 = 4(2b + 1)$. 12. Common denominator is $8b(2b + 1)$. 13. Rewrite fractions: $$\frac{1 + 2b}{4(2b + 1)} = \frac{2b(1 + 2b)}{8b(2b + 1)}$$ $$\frac{2b + 9}{8b} = \frac{(2b + 9)(2b + 1)}{8b(2b + 1)}$$ 14. Subtract numerators: $$2b(1 + 2b) - (2b + 9)(2b + 1)$$ 15. Expand: $$2b + 4b^2 - (4b^2 + 2b + 18b + 9) = 2b + 4b^2 - 4b^2 - 20b - 9 = -18b - 9$$ 16. Factor numerator: $$-9(2b + 1)$$ 17. Final expression: $$\frac{-9(2b + 1)}{8b(2b + 1)} = \frac{\cancel{-9}(\cancel{2b + 1})}{8b(\cancel{2b + 1})} = \frac{-9}{8b}$$ --- 18. **Problem 2.224 c)** Simplify $ \frac{12c^2 + 48}{3} + \frac{96 - 12c^3}{3c - 6} $. 19. Factor denominators and numerators: $3c - 6 = 3(c - 2)$ $12c^2 + 48 = 12(c^2 + 4)$ $96 - 12c^3 = 12(8 - c^3)$ 20. Rewrite: $$\frac{12(c^2 + 4)}{3} + \frac{12(8 - c^3)}{3(c - 2)} = 4(c^2 + 4) + \frac{12(8 - c^3)}{3(c - 2)}$$ 21. Simplify first term: $$4c^2 + 16$$ 22. Factor numerator of second term: $$8 - c^3 = (2)^3 - c^3 = (2 - c)(4 + 2c + c^2)$$ 23. Rewrite second term: $$\frac{12(2 - c)(4 + 2c + c^2)}{3(c - 2)}$$ 24. Note $2 - c = -(c - 2)$, so: $$\frac{12 \cdot -(c - 2)(4 + 2c + c^2)}{3(c - 2)} = \frac{-12 (c - 2)(4 + 2c + c^2)}{3(c - 2)}$$ 25. Cancel $c - 2$: $$\frac{-12 \cancel{(c - 2)} (4 + 2c + c^2)}{3 \cancel{(c - 2)}} = -4(4 + 2c + c^2)$$ 26. Expand: $$-16 - 8c - 4c^2$$ 27. Add both terms: $$(4c^2 + 16) + (-16 - 8c - 4c^2) = 4c^2 + 16 - 16 - 8c - 4c^2 = -8c$$ 28. Final answer: $$-8c$$ --- 29. **Problem 2.225 a)** Simplify $ \frac{5 - 3g}{8g^2 + 2g} + \frac{5g}{12g^2} $. 30. Factor denominator $8g^2 + 2g = 2g(4g + 1)$. 31. Common denominator is $12g^2(4g + 1)$. 32. Rewrite fractions: $$\frac{5 - 3g}{2g(4g + 1)} = \frac{6g(5 - 3g)}{12g^2(4g + 1)}$$ $$\frac{5g}{12g^2} = \frac{5g(4g + 1)}{12g^2(4g + 1)}$$ 33. Add numerators: $$6g(5 - 3g) + 5g(4g + 1) = 30g - 18g^2 + 20g^2 + 5g = (30g + 5g) + (-18g^2 + 20g^2) = 35g + 2g^2$$ 34. Final expression: $$\frac{2g^2 + 35g}{12g^2(4g + 1)}$$ 35. Factor numerator: $$g(2g + 35)$$ 36. Final simplified form: $$\frac{g(2g + 35)}{12g^2(4g + 1)} = \frac{\cancel{g}(2g + 35)}{12g \cdot (4g + 1)} = \frac{2g + 35}{12g(4g + 1)}$$ --- 37. **Problem 2.225 b)** Simplify $ \frac{3 - h}{2h + 6} - \frac{3h^2 + 11}{6h^2} $. 38. Factor denominator $2h + 6 = 2(h + 3)$. 39. Common denominator is $6h^2(h + 3)$. 40. Rewrite fractions: $$\frac{3 - h}{2(h + 3)} = \frac{3h^2(3 - h)}{6h^2(h + 3)}$$ $$\frac{3h^2 + 11}{6h^2} = \frac{(3h^2 + 11)(h + 3)}{6h^2(h + 3)}$$ 41. Subtract numerators: $$3h^2(3 - h) - (3h^2 + 11)(h + 3)$$ 42. Expand: $$9h^2 - 3h^3 - (3h^3 + 9h^2 + 11h + 33) = 9h^2 - 3h^3 - 3h^3 - 9h^2 - 11h - 33 = -6h^3 - 11h - 33$$ 43. Final expression: $$\frac{-6h^3 - 11h - 33}{6h^2(h + 3)}$$ 44. Factor numerator: $$- (6h^3 + 11h + 33)$$ No further factorization obvious. --- 45. **Problem 2.225 c)** Simplify $ \frac{5e + 6}{15e^2} + \frac{6 - e}{3e^2 - 9e} $. 46. Factor denominator $3e^2 - 9e = 3e(e - 3)$. 47. Common denominator is $15e^2(e - 3)$. 48. Rewrite fractions: $$\frac{5e + 6}{15e^2} = \frac{5e + 6}{15e^2} \cdot \frac{e - 3}{e - 3} = \frac{(5e + 6)(e - 3)}{15e^2(e - 3)}$$ $$\frac{6 - e}{3e(e - 3)} = \frac{(6 - e) \cdot 5e}{15e^2(e - 3)} = \frac{5e(6 - e)}{15e^2(e - 3)}$$ 49. Add numerators: $$(5e + 6)(e - 3) + 5e(6 - e)$$ 50. Expand: $$(5e^2 - 15e + 6e - 18) + (30e - 5e^2) = (5e^2 - 15e + 6e - 18) + 30e - 5e^2 = (5e^2 - 5e^2) + (-15e + 6e + 30e) - 18 = 21e - 18$$ 51. Factor numerator: $$3(7e - 6)$$ 52. Final expression: $$\frac{3(7e - 6)}{15e^2(e - 3)} = \frac{\cancel{3}(7e - 6)}{5e^2(e - 3)\cancel{3}} = \frac{7e - 6}{5e^2(e - 3)}$$