`64a^3-343`

Answer

To factor the expression \( 64a^3 - 343 \), we can recognize that it is a difference of cubes. The general formula for factoring a difference of cubes is:

\[ A^3 - B^3 = (A - B)(A^2 + AB + B^2) \]

### Step 1: Identify \( A \) and \( B \)
In our case, we can express \( 64a^3 \) and \( 343 \) as cubes:
- \( 64a^3 = (4a)^3 \)
- \( 343 = 7^3 \)

Thus, we can set:
- \( A = 4a \)
- \( B = 7 \)

### Step 2: Apply the Difference of Cubes Formula
Using the formula, we can write:
\[ 64a^3 - 343 = (4a)^3 - 7^3 = (4a - 7)((4a)^2 + (4a)(7) + 7^2) \]

### Step 3: Calculate Each Part
1. **First Factor**:
\[ 4a - 7 \]

2. **Second Factor**:
- Calculate \( (4a)^2 \):
\[ (4a)^2 = 16a^2 \]
- Calculate \( (4a)(7) \):
\[ (4a)(7) = 28a \]
- Calculate \( 7^2 \):
\[ 7^2 = 49 \]

Putting it all together, the second factor becomes:
\[ 16a^2 + 28a + 49 \]

### Step 4: Write the Final Factored Form
Combining both factors, we have:
\[ 64a^3 - 343 = (4a - 7)(16a^2 + 28a + 49) \]

### Final Answer:
The factored form of \( 64a^3 - 343 \) is:
\[ (4a - 7)(16a^2 + 28a + 49) \]

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