1. Simplify expression 3a + 6b by factoring out the common factor 3: $$3a + 6b = 3(a + 2b)$$ 2. Simplify 4xy – 6yz by factoring out 2y: $$4xy - 6yz = 2y(2x - 3z)$$ 3. Simplify 2u + av – 2v – au by grouping: $$2u - au + av - 2v = u(2 - a) + v(a - 2) = (2 - a)(u - v)$$ 4. Simplify xy + 4x – 2y – 8 by grouping: $$x(y + 4) - 2(y + 4) = (x - 2)(y + 4)$$ 5. Simplify 3x – py – 3y + px by grouping: $$3x + px - py - 3y = x(3 + p) - y(p + 3) = (3 + p)(x - y)$$ 6. Simplify 6xz – 16y – 24x + 4yz by grouping: $$6xz + 4yz - 24x - 16y = 2z(3x + 2y) - 8(3x + 2y) = (3x + 2y)(2z - 8) = 2(3x + 2y)(z - 4)$$ 7. Simplify 15ac – 9ad – 30bc + 18bd by grouping: $$3a(5c - 3d) - 6b(5c - 3d) = (5c - 3d)(3a - 6b) = 3(5c - 3d)(a - 2b)$$ 8. Simplify x² – 16 as a difference of squares: $$x^2 - 16 = (x - 4)(x + 4)$$ 9. Simplify 1 – 4b + 4b² by recognizing a perfect square trinomial: $$1 - 4b + 4b^2 = (1 - 2b)^2$$ 10. Simplify 3t² – 108a² by factoring out 3: $$3(t^2 - 36a^2) = 3(t - 6a)(t + 6a)$$ 11. Simplify x³y – 25xy³ by factoring xy: $$xy(x^2 - 25y^2) = xy(x - 5y)(x + 5y)$$ 12. Simplify x² + x – 2 by factoring: $$(x + 2)(x - 1)$$ 13. Simplify x² – 15x + 54 by factoring: $$(x - 6)(x - 9)$$ 14. Simplify 2x² + 2x – 12 by factoring out 2 and then factoring: $$2(x^2 + x - 6) = 2(x + 3)(x - 2)$$ 15. Simplify 2x² + 5x + 3 by factoring: $$(2x + 3)(x + 1)$$ 16. Simplify 5x² – 17x + 6 by factoring: $$(5x - 2)(x - 3)$$ 17. Simplify 25b⁴ – 120b⁵ + 144b⁶ by factoring out $b^4$ and recognizing a perfect square: $$b^4(25 - 120b + 144b^2) = b^4(5 - 12b)^2$$ 18. Simplify 3q⁴ + 20q² + 32 by factoring (if possible). This may not factor nicely with integers. 19. Simplify x² + 6xy + 5y² by factoring: $$(x + y)(x + 5y)$$ 20. Simplify 4y² – 25 as a difference of squares: $$(2y - 5)(2y + 5)$$ 21. Simplify x⁵ – 4x³y² by factoring out $x^3$: $$x^3(x^2 - 4y^2) = x^3(x - 2y)(x + 2y)$$ 22. Simplify 1 + 18ab + 81a²b² by recognizing a perfect square: $$(1 + 9ab)^2$$ 23. Simplify 144x² – 25y² as a difference of squares: $$(12x - 5y)(12x + 5y)$$ 24. Simplify x² – 7x + 12 by factoring: $$(x - 3)(x - 4)$$ 25. Simplify x² – 14x + 48 by factoring: $$(x - 6)(x - 8)$$ 26. Simplify 3x² – 6x + 3 by factoring out 3: $$3(x^2 - 2x + 1) = 3(x - 1)^2$$ 27. Simplify 6x² + 10x – 4 by factoring out 2: $$2(3x^2 + 5x - 2) = 2(3x - 1)(x + 2)$$ 28. Simplify 1 + 49a² + 14a by rewriting: $$1 + 14a + 49a^2 = (1 + 7a)^2$$ 29. Simplify 2t² – 3t – 14 by factoring: $$(2t + 7)(t - 2)$$ 30. Simplify 9x²y² – 18x³y³ + 9x⁴y⁴ by factoring out $9x^2y^2$: $$9x^2y^2(1 - 2xy + x^2y^2) = 9x^2y^2(1 - xy)^2$$ 31. Simplify 10p² + 3p – 18 by factoring: $$(5p - 6)(2p + 3)$$ 32. Simplify x² – 4xy – 5y² by factoring: $$(x - 5y)(x + y)$$ 33. Simplify 2x² – 9xy + 10y² by factoring: $$(2x - 5y)(x - 2y)$$ 34. Simplify 8t³ + 125 as sum of cubes: $$(2t + 5)(4t^2 - 10t + 25)$$ 35. Simplify 2px – 3y + py – 6x by grouping: $$2px + py - 6x - 3y = p(2x + y) - 3(2x + y) = (p - 3)(2x + y)$$ 36. Simplify pq – 6q – 3p + 18 by grouping: $$q(p - 6) - 3(p - 6) = (p - 6)(q - 3)$$ 37. Simplify x⁴ – 16y⁴ as a difference of squares: $$(x^2 - 4y^2)(x^2 + 4y^2) = (x - 2y)(x + 2y)(x^2 + 4y^2)$$ 38. Simplify 2t² + tu – 6u² by factoring: $$(2t + 3u)(t - 2u)$$ 39. Simplify x² – 81 as a difference of squares: $$(x - 9)(x + 9)$$ 40. Simplify 64x⁴y² – 27xy⁵ by factoring out $xy^2$: $$xy^2(64x^3 - 27y^3) = xy^2(4x - 3y)(16x^2 + 12xy + 9y^2)$$ 41. Simplify $\frac{361}{9} - \frac{114}{a^2} + \frac{81}{a^4}$ by rewriting as: $$\left(\frac{19}{3} - \frac{9}{a^2}\right)^2$$ 42. Simplify x²y² – 9y² – 4x² + 36 by grouping: $$(y^2 - 4)(x^2 - 9) = (y - 2)(y + 2)(x - 3)(x + 3)$$ 43. Simplify x²z² – 4z² + x⁴ – 4x² by grouping: $$(z^2 - 4)(x^2 - 4) = (z - 2)(z + 2)(x - 2)(x + 2)$$ 44. Simplify 169p⁶ – 260p³m² + 100m⁴ by factoring: $$(13p^3 - 10m^2)^2$$ 45. Simplify x³ + y³ + x²y + xy² by grouping: $$(x^3 + y^3) + xy(x + y) = (x + y)(x^2 - xy + y^2) + xy(x + y) = (x + y)(x^2 - xy + y^2 + xy) = (x + y)(x^2 + y^2)$$ 47. Simplify 2(x + y)² + 5(x + y) + 2 by substitution $z = x + y$: $$2z^2 + 5z + 2 = (2z + 1)(z + 2) = (2(x + y) + 1)(x + y + 2)$$ 48. Simplify 2(p – q)² – (p – q) – 1 by substitution $z = p - q$: $$2z^2 - z - 1 = (2z + 1)(z - 1) = (2(p - q) + 1)(p - q - 1)$$ 49. Simplify x⁶ – 8y⁶ as difference of cubes: $$(x^2)^3 - (2y^2)^3 = (x^2 - 2y^2)(x^4 + 2x^2y^2 + 4y^4)$$ 50. Expression x⁴ + 4y⁴ does not factor over the reals easily. 51. Simplify $\frac{25}{36} - 49a^8y^6$ recognizing difference of squares with powers: $$(\frac{5}{6})^2 - (7a^4y^3)^2 = (\frac{5}{6} - 7a^4y^3)(\frac{5}{6} + 7a^4y^3)$$ 52. Expression $(\frac{9}{225})x^2y^6 + (\frac{4}{7})x^3y^6 + (\frac{100}{49})x^4$ can be rewritten but does not factor simply. 53. Simplify 196 + 25x² – 140x by rewriting: $$25x^2 - 140x + 196 = (5x - 14)^2$$ 54. Simplify 121m²n⁶ + 44mn³ + 4 by grouping: $$(11mn^3 + 2)^2$$ 55. Simplify $(m - n)^2 - 2(a - m)(m + n) + (a - m)^2$ by expanding and grouping: $$(m - n)^2 - 2(a - m)(m + n) + (a - m)^2 = (m - n - (a - m))^2 = (2m - n - a)^2$$ 56. Simplify a² – 25 as difference of squares: $$(a - 5)(a + 5)$$ 57. Simplify 4a² – 9 as difference of squares: $$(2a - 3)(2a + 3)$$ 58. Simplify 100 – x²y⁶ as difference of squares: $$(10 - xy^3)(10 + xy^3)$$ Final answers provided for each expression in factorized or simplified form where applicable.