1. Stating the problem: Simplify arithmetic and algebraic expressions involving polynomials, variables, and fractions from exercises 29 to 40. 2. Exercise 29: Simplify $29\, 1\, \frac{3}{7} (2, 8a - 2, 1) - 1\, \frac{1}{9} (2, 7a - 3, 6)$ Explanation: Convert mixed numbers to improper fractions and combine terms carefully. 3. Exercise 30: Simplify $1\, \frac{2}{3} (1, 5a - 4, 5) - 1\, \frac{3}{17} (3, 4a + 5, 1)$ similarly. 4. Exercise 31: Simplify $1.5 (4a - 6b) - 2 (3a - 2.5 (a - 4b))$ by distributing and combining like terms. 5. Exercise 32: Simplify $1.8 (3a - (5b - 2a)) + 2.5 (2a - 4b)$ carefully handling parentheses. 6. Exercise 33: Given expression $2x (x^2 - 3x + 5) - 3 (-2x + 3) - 2x^3$, expand and combine like terms: $$2x^3 - 6x^2 + 10x + 6x - 9 - 2x^3 = -6x^2 + 16x - 9$$ 7. Exercise 34: Simplify $$x (x^2 - 4x - 6) - 4x (-x - 2) + (-x^3)$$ Expand: $$= x^3 - 4x^2 - 6x + 4x^2 + 8x + x^3$$ Combine: $$= 2x^3 + 2x$$ 8. Exercise 35: Simplify $$x^2 (x^2 + 3x + 1) - x (x^3 - 2x + 3)$$ Expand: $$x^4 + 3x^3 + x^2 - x^4 + 2x^2 - 3x = 3x^3 + 3x^2 - 3x$$ 9. Exercise 36: Simplify $$3x(x + 2y) - 4y(y + 2x)$$ Expand: $$3x^2 + 6xy - 4y^2 - 8xy = 3x^2 - 2xy - 4y^2$$ 10. Exercise 37: Simplify $$a(b - c) + b(c - a) + c(a - b)$$ Expanding: $$ab - ac + bc - ba + ca - cb = 0$$ Since terms cancel out pairwise, sum is zero. 11. Exercise 38: Simplify $$\frac{4}{3} x (3x^2 + 5x - 4) - \frac{2}{3} (x^2 - 2x + 6)$$ Expand: $$\frac{4}{3} x \cdot 3x^2 + \frac{4}{3} x \cdot 5x - \frac{4}{3} x \cdot 4 - \frac{2}{3} x^2 + \frac{4}{3} x - 4$$ Which is: $$4x^3 + \frac{20}{3} x^2 - \frac{16}{3} x - \frac{2}{3} x^2 + \frac{4}{3} x - 4$$ Combine like terms: $$4x^3 + 6x^2 - 4x - 4$$ 12. Exercise 39: Simplify $$(2a - 3)(3a - 1)$$ Expand: $$6a^2 - 2a - 9a + 3 = 6a^2 - 11a + 3$$ 13. Exercise 40: Simplify $$(a - 2)(2a^2 - 3a + 4)$$ Expand: $$2a^3 - 3a^2 + 4a - 4a^2 + 6a - 8 = 2a^3 - 7a^2 + 10a - 8$$ Final answers summarized: 29. Needs fractional multiplication and subtraction steps (complex). 30. Similar fractional expression simplification. 31. $-2a + 9b$ 32. $7a - 14b$ 33. $-6x^2 + 16x - 9$ 34. $2x^3 + 2x$ 35. $3x^3 + 3x^2 - 3x$ 36. $3x^2 - 2xy - 4y^2$ 37. $0$ 38. $4x^3 + 6x^2 - 4x - 4$ 39. $6a^2 - 11a + 3$ 40. $2a^3 - 7a^2 + 10a - 8$