`(4(sqrt(6) + sqrt(2)))/(sqrt(6) - sqrt(2)) - (2 + sqrt(3))/(2 - sqrt(3))` =

Options

A. 1
B. `sqrt(6) - sqrt(2)`
C. `sqrt(6) + sqrt(2)`
D. 8

Answer

To solve the expression \((4(\sqrt{6} + \sqrt{2})) / (\sqrt{6} - \sqrt{2}) - (2 + \sqrt{3}) / (2 - \sqrt{3})\), we will simplify each part separately and then combine the results.

### Step 1: Simplify the first expression
We start with the first expression:

\[ \frac{4(\sqrt{6} + \sqrt{2})}{\sqrt{6} - \sqrt{2}} \]

To simplify this, we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \((\sqrt{6} + \sqrt{2})\):

\[ \frac{4(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} \]

### Step 2: Calculate the denominator
Now, we calculate the denominator using the difference of squares:

\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 \]

### Step 3: Calculate the numerator
Next, we calculate the numerator:

\[ 4(\sqrt{6} + \sqrt{2})^2 = 4(\sqrt{6}^2 + 2\sqrt{6}\sqrt{2} + \sqrt{2}^2) = 4(6 + 2\sqrt{12} + 2) = 4(8 + 2\sqrt{12}) = 32 + 8\sqrt{12} \]

### Step 4: Combine the results
Now we can combine the results:

\[ \frac{32 + 8\sqrt{12}}{4} = 8 + 2\sqrt{12} \]

### Step 5: Simplify \(\sqrt{12}\)
We can simplify \(\sqrt{12}\):

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

So, we substitute back:

\[ 8 + 2(2\sqrt{3}) = 8 + 4\sqrt{3} \]

### Step 6: Simplify the second expression
Now we simplify the second expression:

\[ \frac{2 + \sqrt{3}}{2 - \sqrt{3}} \]

Again, we will rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is \((2 + \sqrt{3})\):

\[ \frac{(2 + \sqrt{3})(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} \]

### Step 7: Calculate the denominator
Calculating the denominator:

\[ (2)^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

### Step 8: Calculate the numerator
Now, calculating the numerator:

\[ (2 + \sqrt{3})^2 = 2^2 + 2(2)(\sqrt{3}) + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

### Step 9: Combine the results
So, the second expression simplifies to:

\[ 7 + 4\sqrt{3} \]

### Step 10: Subtract the second expression from the first
Now, we subtract the second expression from the first:

\[ (8 + 4\sqrt{3}) - (7 + 4\sqrt{3}) = 8 - 7 + 4\sqrt{3} - 4\sqrt{3} = 1 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{1} \]