If a, b, c, d satisfy the equation `a + 7b + 3c + 5d = 0`, `8a + 4b + 6c + 2d = - 16`, `2a + 6b + 4c + 8d =16` and `5a + 3b + 7c + d = -16` then the value of `(a + d) (b + c)` =______

Answer

To solve the problem, we need to find the value of \((a + d)(b + c)\) given the equations:

1. \(a + 7b + 3c + 5d = 0\) (Equation 1)
2. \(8a + 4b + 6c + 2d = -16\) (Equation 2)
3. \(2a + 6b + 4c + 8d = 16\) (Equation 3)
4. \(5a + 3b + 7c + d = -16\) (Equation 4)

### Step 1: Add Equation 1 and Equation 4
We start by adding Equation 1 and Equation 4.

\[ (a + 7b + 3c + 5d) + (5a + 3b + 7c + d) = 0 - 16 \]

This simplifies to:

\[ 6a + 10b + 10c + 6d = -16 \]

### Step 2: Factor out common terms
We can factor out common terms from the equation:

\[ 6(a + d) + 10(b + c) = -16 \]

### Step 3: Add Equation 2 and Equation 3
Next, we add Equation 2 and Equation 3.

\[ (8a + 4b + 6c + 2d) + (2a + 6b + 4c + 8d) = -16 + 16 \]

This simplifies to:

\[ 10a + 10b + 10c + 10d = 0 \]

### Step 4: Factor out common terms
We can factor out common terms from this equation:

\[ 10(a + b + c + d) = 0 \]

This implies:

\[ a + b + c + d = 0 \]

### Step 5: Express \(d\) in terms of \(a\), \(b\), and \(c\)
From the equation \(a + b + c + d = 0\), we can express \(d\) as:

\[ d = - (a + b + c) \]

### Step 6: Substitute \(d\) into the equation from Step 2
Now, substituting \(d\) into the equation from Step 2:

\[ 6(a - (a + b + c)) + 10(b + c) = -16 \]

This simplifies to:

\[ 6(-b - c) + 10(b + c) = -16 \]

### Step 7: Combine like terms
Combining the terms gives:

\[ (-6b - 6c + 10b + 10c) = -16 \]

This simplifies to:

\[ 4b + 4c = -16 \]

### Step 8: Solve for \(b + c\)
Dividing the entire equation by 4:

\[ b + c = -4 \]

### Step 9: Substitute \(b + c\) back to find \(a + d\)
Now substituting \(b + c = -4\) back into the equation from Step 2:

\[ 6(a + d) + 10(-4) = -16 \]

This simplifies to:

\[ 6(a + d) - 40 = -16 \]

### Step 10: Solve for \(a + d\)
Adding 40 to both sides:

\[ 6(a + d) = 24 \]

Dividing by 6:

\[ a + d = 4 \]

### Step 11: Calculate \((a + d)(b + c)\)
Now we can find \((a + d)(b + c)\):

\[ (a + d)(b + c) = 4 \times (-4) = -16 \]

Thus, the final answer is:

\[ \boxed{-16} \]