The factor form `(2x+3y)^(3)-(2x-3y)^(3)` is

Options

A. `18y(4x^(2)+3y^(2))`
B. `18y(4x-3y)`
C. `18y^(2)(4x+3y)`
D. `18y(4x^(2)-3y^(2))`

Answer

To factor the expression \((2x + 3y)^3 - (2x - 3y)^3\), we can use the formula for the difference of cubes, which is given by:

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

where \(A = 2x + 3y\) and \(B = 2x - 3y\).

### Step 1: Identify A and B
Let:
- \(A = 2x + 3y\)
- \(B = 2x - 3y\)

### Step 2: Apply the difference of cubes formula
Using the difference of cubes formula, we have:

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

### Step 3: Calculate \(A - B\)
Now we calculate \(A - B\):

\[ A - B = (2x + 3y) - (2x - 3y) = 2x + 3y - 2x + 3y = 6y \]

### Step 4: Calculate \(A^2\), \(AB\), and \(B^2\)
Next, we need to calculate \(A^2\), \(AB\), and \(B^2\):

1. **Calculate \(A^2\)**:
\[ A^2 = (2x + 3y)^2 = 4x^2 + 12xy + 9y^2 \]

2. **Calculate \(B^2\)**:
\[ B^2 = (2x - 3y)^2 = 4x^2 - 12xy + 9y^2 \]

3. **Calculate \(AB\)**:
\[ AB = (2x + 3y)(2x - 3y) = 4x^2 - 9y^2 \]

### Step 5: Combine \(A^2\), \(AB\), and \(B^2\)
Now we combine these results:

\[ A^2 + AB + B^2 = (4x^2 + 12xy + 9y^2) + (4x^2 - 9y^2) + (4x^2 - 12xy + 9y^2) \]

Combine like terms:

\[ = 4x^2 + 4x^2 + 4x^2 + 12xy - 12xy + 9y^2 - 9y^2 = 12x^2 \]

### Step 6: Final expression
Now substituting back into the expression we have:

\[ (2x + 3y)^3 - (2x - 3y)^3 = (6y)(12x^2) = 72xy \]

### Final Answer
Thus, the factor form of \((2x + 3y)^3 - (2x - 3y)^3\) is:

\[ 72y(2x^2) \]