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

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 expand both sides and simplify.

### Step 1: Expand both sides

**Left Side:**
\[ (a - 2b - 3c + 4d)(a + 2b + 3c + 4d) \]
Using the distributive property (FOIL method):
\[ = a^2 + 2ab + 3ac + 4ad - 2ab - 4b^2 - 6bc - 8bd - 3ac - 6bc - 9c^2 - 12cd + 4ad + 8bd + 12cd + 16d^2 \]
Combining like terms:
\[ = a^2 + 0 + 0 + 0 - 4b^2 - 9c^2 + 16d^2 \]

**Right Side:**
\[ (a + 2b - 3c - 4d)(a - 2b + 3c - 4d) \]
Using the distributive property:
\[ = a^2 - 2ab + 3ac - 4ad + 2ab - 4b^2 - 6bc + 8bd - 3ac + 6bc - 9c^2 + 12cd - 4ad + 8bd - 12cd + 16d^2 \]
Combining like terms:
\[ = a^2 + 0 + 0 + 0 - 4b^2 - 9c^2 + 16d^2 \]

### Step 2: Set both sides equal

From the expansions, we have:
\[ a^2 - 4b^2 - 9c^2 + 16d^2 = a^2 - 4b^2 - 9c^2 + 16d^2 \]
This shows that both sides are equal, confirming our calculations.

### Step 3: Find \(2ad\)

Since the equation holds true, we can derive \(2ad\) directly from the terms involving \(d\):
From the original equation, we can isolate \(2ad\):
\[ 2ad = 0 \]

### Final Answer:
\[ 2ad = 0 \]

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