Verify that `(11 pq + 4q) ^(2) - (11 pq -4q) ^(2) = 17pq ^(2)`

Answer

To verify the expression \((11pq + 4q)^2 - (11pq - 4q)^2 = 17pq^2\), we will use the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\).

### Step 1: Identify \(a\) and \(b\)
Let:
- \(a = 11pq + 4q\)
- \(b = 11pq - 4q\)

### Step 2: Apply the difference of squares formula
Using the difference of squares formula:
\[ (11pq + 4q)^2 - (11pq - 4q)^2 = (a - b)(a + b) \]

### Step 3: Calculate \(a - b\)
\[ a - b = (11pq + 4q) - (11pq - 4q) = 11pq + 4q - 11pq + 4q = 8q \]

### Step 4: Calculate \(a + b\)
\[ a + b = (11pq + 4q) + (11pq - 4q) = 11pq + 4q + 11pq - 4q = 22pq \]

### Step 5: Substitute back into the difference of squares formula
Now substituting \(a - b\) and \(a + b\) into the formula:
\[ (11pq + 4q)^2 - (11pq - 4q)^2 = (8q)(22pq) \]

### Step 6: Simplify the expression
Now we simplify:
\[ (8q)(22pq) = 176pq^2 \]

### Step 7: Conclusion
We have shown that:
\[ (11pq + 4q)^2 - (11pq - 4q)^2 = 176pq^2 \]
However, the original expression we needed to verify was \(17pq^2\). Thus, we conclude that:
\[ (11pq + 4q)^2 - (11pq - 4q)^2 \neq 17pq^2 \]