For the expansion of $(2x + 1)^3$ and $(2x - 2)^3$, if all terms are correct with or without signs (and no additional terms), 3 or 4 terms correct with signs, $$ (2x + 1)^3 = 8x^3 + 12x^2 + 6x + 1 $$ or for correct expression after the expansion: $$ (2x - 2)^3 = 8x^3 - 12x^2 + 12x - 8 $$ or for correct expression after expansion: $$ = (2x)^3 - 3(2x)^2(2) + 3(2x)(2^2) - (2)^3 $$ or $(x^2 - 2)(x^2 + 2)$ where relevant - Edexcel - GCSE Maths - Question 16 - 2022 - Paper 1

To find the expansion

1. Utilize the binomial theorem: $(a + b)^n = \sum_{k=0}^{n} C(n, k) a^{n-k} b^k$ where $C(n, k)$ is the binomial coefficient.

2. For the expansion of $(2x + 1)^3$: $= C(3, 0) (2x)^3 (1)^0 + C(3, 1) (2x)^2 (1)^1 + C(3, 2) (2x)^1 (1)^2 + C(3, 3) (1)^3$ $= 8x^3 + 12x^2 + 6x + 1$

3. For $(2x - 2)^3$: $= C(3, 0) (2x)^3 (-2)^0 + C(3, 1) (2x)^2 (-2)^1 + C(3, 2) (2x)^1 (-2)^2 + C(3, 3) (-2)^3$ $= 8x^3 - 12x^2 + 12x - 8$

Thus, both expansions are provided correctly.