To solve the problem of expanding $(2x - 7)^3$, we will use the binomial expansion formula:

$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

Let $a = 2x$ and $b = 7$.

Step-by-step expansion:

1. $a^3 = (2x)^3 = 8x^3$
2. $3a^2b = 3(2x)^2(7) = 3(4x^2)(7) = 84x^2$
3. $3ab^2 = 3(2x)(7^2) = 3(2x)(49) = 294x$
4. $b^3 = 7^3 = 343$

Putting it all together: $(2x - 7)^3 = 8x^3 - 84x^2 + 294x - 343$

The correct answer is D: $8x^3 - 84x^2 + 294x - 343$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How is the binomial expansion formula derived?
2. Can the binomial expansion be applied to powers higher than 3?
3. What are some practical applications of the binomial expansion in real life?
4. How does expanding $(2x + 7)^3$ differ from $(2x - 7)^3$?
5. What happens if you expand $(2x - 7)^4$?

Tip: Always check your signs carefully when expanding binomials, especially with negative terms.