Expand `: (x-2y + 6)( x-2y -6) .`

Answer

To expand the expression \((x - 2y + 6)(x - 2y - 6)\), we can use the difference of squares formula, which states that \(a^2 - b^2 = (a + b)(a - b)\).

### Step-by-Step Solution:

1. **Identify the common terms**:
We can see that both terms in the expression share a common factor of \((x - 2y)\). We can denote this common term as \(a\) and the constant \(6\) as \(b\). Thus, we can rewrite the expression as:
\[ (a + b)(a - b) \quad \text{where } a = (x - 2y) \text{ and } b = 6. \]

2. **Apply the difference of squares formula**:
Using the formula \(a^2 - b^2\), we can expand the expression:
\[ (x - 2y + 6)(x - 2y - 6) = (x - 2y)^2 - 6^2. \]

3. **Calculate \((x - 2y)^2\)**:
To expand \((x - 2y)^2\), we use the formula \((a - b)^2 = a^2 - 2ab + b^2\):
\[ (x - 2y)^2 = x^2 - 2 \cdot x \cdot 2y + (2y)^2 = x^2 - 4xy + 4y^2. \]

4. **Calculate \(6^2\)**:
\[ 6^2 = 36. \]

5. **Combine the results**:
Now substituting back into our expression:
\[ (x - 2y + 6)(x - 2y - 6) = (x^2 - 4xy + 4y^2) - 36. \]

6. **Final expression**:
Thus, the expanded form of the expression is:
\[ x^2 - 4xy + 4y^2 - 36. \]

### Final Answer:
\[ x^2 - 4xy + 4y^2 - 36. \]