To solve this question, we use the Binomial Theorem, which states:

$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$

Here, the given binomial is $(4a - b)^3$. We will expand this step-by-step.

Step 1: Identify components

• $a = 4a$
• $b = -b$
• $n = 3$

The general term of the expansion is:

$\binom{3}{k} (4a)^{3-k} (-b)^k$

Step 2: Calculate each term

For $k = 0, 1, 2, 3$:

1. When $k = 0$: $\binom{3}{0} (4a)^{3-0} (-b)^0 = 1 \cdot (4a)^3 \cdot 1 = 64a^3$

2. When $k = 1$: $\binom{3}{1} (4a)^{3-1} (-b)^1 = 3 \cdot (4a)^2 \cdot (-b) = 3 \cdot 16a^2 \cdot (-b) = -48a^2b$

3. When $k = 2$: $\binom{3}{2} (4a)^{3-2} (-b)^2 = 3 \cdot (4a)^1 \cdot b^2 = 3 \cdot 4a \cdot b^2 = 12ab^2$

4. When $k = 3$: $\binom{3}{3} (4a)^{3-3} (-b)^3 = 1 \cdot 1 \cdot (-b)^3 = -b^3$

Step 3: Combine the terms

The expansion of $(4a - b)^3$ is:

$64a^3 - 48a^2b + 12ab^2 - b^3$

Step 4: Match the terms to the blanks

From the provided choices:

1. $64a^3$
2. $-48a^2b$ → $48a^2b$ (negative sign is included)
3. $12ab^2$
4. $-b^3$

The correct arrangement is:

[ \text{[64a³]} - \text{[48a²b]} + \text{[12ab²]} - \text{[b³]} ]

Final Answer:

• First box: 64a³
• Second box: 48a²b
• Third box: 12ab²
• Fourth box: b³

Let me know if you'd like me to explain any step in more detail!

Related Questions:

1. What is the binomial theorem, and how is it applied for powers of 4?
2. How do you compute the binomial coefficients $\binom{n}{k}$?
3. How do you expand a binomial with a negative term?
4. What happens when the power $n$ increases in a binomial expansion?
5. What is the relationship between Pascal's Triangle and the binomial theorem?

Tip: Always double-check the powers and coefficients of each term when expanding a binomial!