show that `(3a+2b-c+d)^(2)-12a(2b-c+d)` is a perfect square .

Answer

To show that the expression \((3a + 2b - c + d)^2 - 12a(2b - c + d)\) is a perfect square, we will follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ (3a + 2b - c + d)^2 - 12a(2b - c + d) \]

### Step 2: Identify the first term as a square
The first term \((3a + 2b - c + d)^2\) is already a perfect square.

### Step 3: Rewrite the second term
The second term can be rewritten as:
\[ 12a(2b - c + d) = 2 \cdot 6a(2b - c + d) \]
This helps us identify the coefficients for the perfect square formula.

### Step 4: Compare with the perfect square formula
Recall the formula for a perfect square:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
Here, we can let:
- \(a = 3a\)
- \(b = (2b - c + d)\)

### Step 5: Calculate \(2ab\)
Now, we calculate \(2ab\):
\[ 2ab = 2 \cdot (3a) \cdot (2b - c + d) = 6a(2b - c + d) \]
We see that:
\[ -12a(2b - c + d) = -2 \cdot 6a(2b - c + d) \]

### Step 6: Substitute back into the expression
Substituting back, we have:
\[ (3a + 2b - c + d)^2 - 2 \cdot 6a(2b - c + d) \]

### Step 7: Recognize the perfect square
Now we can recognize that:
\[ (3a + 2b - c + d)^2 - 2 \cdot 6a(2b - c + d) = (3a - (2b - c + d))^2 \]

### Step 8: Final expression
Thus, we can write:
\[ (3a - (2b - c + d))^2 \]
This shows that the original expression is indeed a perfect square.

### Conclusion
Therefore, we have shown that:
\[ (3a + 2b - c + d)^2 - 12a(2b - c + d) = (3a - (2b - c + d))^2 \]
is a perfect square.

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