The positive square root of
`((a + b) ^(2) - (c + d) ^(2))/( (a + b) ^(2) - (c - d ) ^(2)) xx ((a + b + c) ^(2) - d ^(2))/( (a + b -c ) ^(2 ) - d ^(2))` using factorisation is:

Options

A. `(a + b + c + d)/( a + b - c + d)`
B. `(a + b + c- d)/( a + b + c + d)`
C. `(a + b - c + d)/( a + b + c-d)`
D. `(a - b - c + d)/( a + b + c + d)`

Answer

To solve the given expression step by step, we will use the difference of squares formula and factorization techniques. The expression we need to simplify is:

\[ \sqrt{\frac{(a + b)^2 - (c + d)^2}{(a + b)^2 - (c - d)^2}} \times \frac{(a + b + c)^2 - d^2}{(a + b - c)^2 - d^2} \]

### Step 1: Apply the Difference of Squares Formula
Recall the difference of squares formula:
\[ x^2 - y^2 = (x - y)(x + y) \]
We will apply this to both the numerator and denominator of the first fraction.

1. For the first term in the numerator:
\[ (a + b)^2 - (c + d)^2 = ((a + b) - (c + d))((a + b) + (c + d)) = (a + b - c - d)(a + b + c + d) \]

2. For the first term in the denominator:
\[ (a + b)^2 - (c - d)^2 = ((a + b) - (c - d))((a + b) + (c - d)) = (a + b - c + d)(a + b + c - d) \]

### Step 2: Substitute Back into the Expression
Now, substitute these factorizations back into the expression:

\[ \sqrt{\frac{(a + b - c - d)(a + b + c + d)}{(a + b - c + d)(a + b + c - d)}} \times \frac{(a + b + c)^2 - d^2}{(a + b - c)^2 - d^2} \]

### Step 3: Apply the Difference of Squares Again
Now, we will apply the difference of squares formula to the second fraction:

1. For the numerator:
\[ (a + b + c)^2 - d^2 = ((a + b + c) - d)((a + b + c) + d) = (a + b + c - d)(a + b + c + d) \]

2. For the denominator:
\[ (a + b - c)^2 - d^2 = ((a + b - c) - d)((a + b - c) + d) = (a + b - c - d)(a + b - c + d) \]

### Step 4: Substitute Back Again
Now substitute these factorizations into the second fraction:

\[ \frac{(a + b + c - d)(a + b + c + d)}{(a + b - c - d)(a + b - c + d)} \]

### Step 5: Combine the Two Parts
Now we combine everything back together:

\[ \sqrt{\frac{(a + b - c - d)(a + b + c + d)}{(a + b - c + d)(a + b + c - d)}} \times \frac{(a + b + c - d)(a + b + c + d)}{(a + b - c - d)(a + b - c + d)} \]

### Step 6: Simplify the Expression
Notice that \( (a + b - c - d) \) cancels out:

\[ \sqrt{\frac{(a + b + c + d)(a + b + c - d)}{(a + b - c + d)(a + b - c + d)}} \]

### Step 7: Final Simplification
Now we can take the square root:

\[ \frac{(a + b + c + d)}{(a + b - c + d)} \]

### Final Answer
Thus, the positive square root of the given expression is:

\[ \frac{(a + b + c + d)}{(a + b - c + d)} \]