If `(a-2b-3c+4d)(a+2b+3c+4d) = (a+2b-3c-4d)(a-2b+3c-4d)` then `2ad =`
`"(a) 3bc (b) bc (c) 5bc (d) 2bc"`

Options

A. `3bc`
B. `bc`
C. `5bc`
D. `2bc`

Explanation

To solve the equation \((a-2b-3c+4d)(a+2b+3c+4d) = (a+2b-3c-4d)(a-2b+3c-4d)\), we will follow these steps:

### Step 1: Expand both sides of the equation
We will expand both sides of the equation using the distributive property (also known as the FOIL method for binomials).

**Left Side:**
\[ (a - 2b - 3c + 4d)(a + 2b + 3c + 4d) = a^2 + 2ab + 3ac + 4ad - 2ab - 4b^2 - 6bc - 8bd - 3ac - 6bc - 9c^2 - 12cd + 4ad + 8bd + 12cd + 16d^2 \]
Combining like terms:
\[ = a^2 - 4b^2 - 9c^2 + 16d^2 + 8ad - 12bc \]

**Right Side:**
\[ (a + 2b - 3c - 4d)(a - 2b + 3c - 4d) = a^2 - 2ab + 3ac - 4ad + 2ab - 4b^2 - 6bc + 8bd - 3ac + 6bc - 9c^2 + 12cd - 4ad - 8bd + 16d^2 \]
Combining like terms:
\[ = a^2 - 4b^2 - 9c^2 + 16d^2 - 8ad + 12bc \]

### Step 2: Set the expanded forms equal to each other
Now we have:
\[ a^2 - 4b^2 - 9c^2 + 16d^2 + 8ad - 12bc = a^2 - 4b^2 - 9c^2 + 16d^2 - 8ad + 12bc \]

### Step 3: Simplify the equation
Subtract \(a^2 - 4b^2 - 9c^2 + 16d^2\) from both sides:
\[ 8ad - 12bc = -8ad + 12bc \]

### Step 4: Combine like terms
Now, we will add \(8ad\) and \(12bc\) to both sides:
\[ 8ad + 8ad = 12bc + 12bc \]
This simplifies to:
\[ 16ad = 24bc \]

### Step 5: Solve for \(2ad\)
Now, divide both sides by 8:
\[ 2ad = \frac{24bc}{8} = 3bc \]

### Conclusion
Thus, the final answer is:
\[ \boxed{3bc} \]