The set of values of 'c' for which the equation `x^(2)-4x-c-sqrt(8x^(2)-32x-8c)=0` has exactly two distinct real solutions, is (a,b) then find the value of (b-a).

Answer

To solve the equation \( x^2 - 4x - c - \sqrt{8x^2 - 32x - 8c} = 0 \) for the values of \( c \) that yield exactly two distinct real solutions, we will follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ x^2 - 4x - c - \sqrt{8x^2 - 32x - 8c} = 0 \]
We can isolate the square root:
\[ \sqrt{8x^2 - 32x - 8c} = x^2 - 4x - c \]

### Step 2: Squaring Both Sides
Next, we square both sides to eliminate the square root:
\[ 8x^2 - 32x - 8c = (x^2 - 4x - c)^2 \]

### Step 3: Expanding the Right Side
Now we expand the right side:
\[ (x^2 - 4x - c)^2 = x^4 - 8x^3 + (16 + 2c)x^2 + 8cx + c^2 \]
Thus, we have:
\[ 8x^2 - 32x - 8c = x^4 - 8x^3 + (16 + 2c)x^2 + 8cx + c^2 \]

### Step 4: Rearranging into a Polynomial
Rearranging gives us a polynomial:
\[ x^4 - 8x^3 + (16 + 2c - 8)x^2 + (8c + 32)x + (c^2 + 8c) = 0 \]
This simplifies to:
\[ x^4 - 8x^3 + (2c + 8)x^2 + (8c + 32)x + (c^2 + 8c) = 0 \]

### Step 5: Finding Conditions for Distinct Real Solutions
For the polynomial to have exactly two distinct real solutions, we need to consider the conditions on the discriminant. The discriminant of a polynomial must be positive for it to have distinct real roots.

1. The quadratic \( x^2 - 4x - c = 0 \) must have a non-negative discriminant:
\[ D_1 = (-4)^2 - 4(1)(-c) = 16 + 4c \geq 0 \implies c \geq -4 \]

2. The quadratic \( x^2 - 4x - c = \sqrt{8} \) must also yield real solutions:
\[ D_2 = (-4)^2 - 4(1)(-c - \sqrt{8}) = 16 + 4c + 32 \geq 0 \implies c \geq -12 \]

### Step 6: Combining Conditions
From the above conditions, we have:
- \( c \geq -4 \)
- \( c \geq -12 \)

Thus, the values of \( c \) for which the original equation has exactly two distinct real solutions lie in the interval:
\[ -12 < c < -4 \]

### Step 7: Finding \( b - a \)
Here, \( a = -12 \) and \( b = -4 \). Therefore:
\[ b - a = -4 - (-12) = -4 + 12 = 8 \]

### Final Answer
The value of \( b - a \) is:
\[ \boxed{8} \]