Expand: `(3x- 2y + 4)(3x- 2y - 4)`

Answer

To expand the expression \((3x - 2y + 4)(3x - 2y - 4)\), we can use the difference of squares formula. The expression can be rewritten in the form of \((a + b)(a - b)\), where:

- \(a = 3x - 2y\)
- \(b = 4\)

### Step 1: Identify \(a\) and \(b\)
We identify \(a\) and \(b\):
- \(a = 3x - 2y\)
- \(b = 4\)

### Step 2: Apply the difference of squares formula
Using the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\), we can expand the expression:

\[ (3x - 2y + 4)(3x - 2y - 4) = (3x - 2y)^2 - 4^2 \]

### Step 3: Calculate \(a^2\) and \(b^2\)
Now we calculate \(a^2\) and \(b^2\):
- \(a^2 = (3x - 2y)^2\)
- \(b^2 = 4^2 = 16\)

### Step 4: Expand \(a^2\)
To expand \(a^2 = (3x - 2y)^2\), we use the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[ (3x - 2y)^2 = (3x)^2 - 2(3x)(2y) + (2y)^2 \]
Calculating each term:
- \((3x)^2 = 9x^2\)
- \(-2(3x)(2y) = -12xy\)
- \((2y)^2 = 4y^2\)

Thus,
\[ (3x - 2y)^2 = 9x^2 - 12xy + 4y^2 \]

### Step 5: Combine the results
Now we can substitute back into our expression:
\[ (3x - 2y + 4)(3x - 2y - 4) = (9x^2 - 12xy + 4y^2) - 16 \]

### Step 6: Final expression
Combining the terms gives us:
\[ 9x^2 - 12xy + 4y^2 - 16 \]

### Final Answer
The expanded form of \((3x - 2y + 4)(3x - 2y - 4)\) is:
\[ 9x^2 - 12xy + 4y^2 - 16 \]