The question asks to "Remove the brackets and simplify" for each expression. Let's start by expanding the brackets for each expression step by step.

1st row:

1. $(x + 2)(x - 3)(x - 4)$:

   • First, expand $(x + 2)(x - 3)$: $(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6$
   • Now, expand with $(x - 4)$: $(x^2 - x - 6)(x - 4) = x^3 - 4x^2 - x^2 + 4x - 6x + 24 = x^3 - 5x^2 - 2x + 24$

2. $(2x - 1)(x + 1)(x - 1)$:

   • First, expand $(x + 1)(x - 1)$: $(x + 1)(x - 1) = x^2 - 1$
   • Now, expand with $(2x - 1)$: $(2x - 1)(x^2 - 1) = 2x^3 - 2x - x^2 + 1 = 2x^3 - x^2 - 2x + 1$

3. $(6x - 5)(2x + 7)(3x - 8)$:

   • First, expand $(6x - 5)(2x + 7)$: $(6x - 5)(2x + 7) = 12x^2 + 42x - 10x - 35 = 12x^2 + 32x - 35$
   • Now, expand with $(3x - 8)$: $(12x^2 + 32x - 35)(3x - 8) = 36x^3 - 96x^2 + 96x^2 - 256x - 105x + 280 = 36x^3 - 361x + 280$

2nd row:

4. $(x - 1)(2x + 3)^2$:

   • Expand $(2x + 3)^2$: $(2x + 3)^2 = 4x^2 + 12x + 9$
   • Now, expand with $(x - 1)$: $(x - 1)(4x^2 + 12x + 9) = x(4x^2 + 12x + 9) - (4x^2 + 12x + 9) = 4x^3 + 12x^2 + 9x - 4x^2 - 12x - 9 = 4x^3 + 8x^2 - 3x - 9$

5. $(x - 2)^3 - (x + 1)^3$:

   • Use binomial expansion for both cubes: $(x - 2)^3 = x^3 - 6x^2 + 12x - 8$ $(x + 1)^3 = x^3 + 3x^2 + 3x + 1$
   • Subtract the two: $(x - 2)^3 - (x + 1)^3 = (x^3 - 6x^2 + 12x - 8) - (x^3 + 3x^2 + 3x + 1)$ $= x^3 - 6x^2 + 12x - 8 - x^3 - 3x^2 - 3x - 1 = -9x^2 + 9x - 9$

3rd row:

6. $(x - 1)(x - 3)(x - 5)$:

   • First, expand $(x - 1)(x - 3)$: $(x - 1)(x - 3) = x^2 - 3x - x + 3 = x^2 - 4x + 3$
   • Now, expand with $(x - 5)$: $(x^2 - 4x + 3)(x - 5) = x^3 - 5x^2 - 4x^2 + 20x + 3x - 15 = x^3 - 9x^2 + 23x - 15$

7. $(3x + 1)(2x + 1)(x - 2)$:

   • First, expand $(3x + 1)(2x + 1)$: $(3x + 1)(2x + 1) = 6x^2 + 3x + 2x + 1 = 6x^2 + 5x + 1$
   • Now, expand with $(x - 2)$: $(6x^2 + 5x + 1)(x - 2) = 6x^3 - 12x^2 + 5x^2 - 10x + x - 2 = 6x^3 - 7x^2 - 9x - 2$

Do you need more details or want me to continue? Here are some related problems:

1. How do you handle the expansion of cubic expressions?
2. What are some shortcuts for expanding binomials?
3. Can you use special products to expand quadratic terms faster?
4. How does the order of multiplication affect expansion results?
5. Are there any techniques to factorize after expanding?

Tip: Always simplify intermediate steps before proceeding to avoid unnecessary complexity.