`(sqrt(x)+sqrt(y))^(2)=x+y+2sqrt(xy)` and `sqrt(x)sqrt(y)=sqrt(xy)` , where `x` and `y` are positive real numbers .
If `a=1+sqrt(2)+sqrt(3)` and `b=1+sqrt(2)-sqrt(3)` , then `a^(2)+b^(2)-2a-2b=`

Options

A. `11`
B. `8`
C. `152`
D. `15`

Answer

To solve the problem, we need to find the value of \( a^2 + b^2 - 2a - 2b \) given:

- \( a = 1 + \sqrt{2} + \sqrt{3} \)
- \( b = 1 + \sqrt{2} - \sqrt{3} \)

### Step 1: Calculate \( a^2 \)

First, we will calculate \( a^2 \):

\[ a^2 = (1 + \sqrt{2} + \sqrt{3})^2 \]

Using the expansion formula \( (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz \):

\[ a^2 = 1^2 + (\sqrt{2})^2 + (\sqrt{3})^2 + 2(1)(\sqrt{2}) + 2(1)(\sqrt{3}) + 2(\sqrt{2})(\sqrt{3}) \]

Calculating each term:

- \( 1^2 = 1 \)
- \( (\sqrt{2})^2 = 2 \)
- \( (\sqrt{3})^2 = 3 \)
- \( 2(1)(\sqrt{2}) = 2\sqrt{2} \)
- \( 2(1)(\sqrt{3}) = 2\sqrt{3} \)
- \( 2(\sqrt{2})(\sqrt{3}) = 2\sqrt{6} \)

Putting it all together:

\[ a^2 = 1 + 2 + 3 + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{6} = 6 + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{6} \]

### Step 2: Calculate \( b^2 \)

Now, we calculate \( b^2 \):

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

Using the same expansion formula:

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

Calculating each term:

- \( 1^2 = 1 \)
- \( (\sqrt{2})^2 = 2 \)
- \( (-\sqrt{3})^2 = 3 \)
- \( 2(1)(\sqrt{2}) = 2\sqrt{2} \)
- \( 2(1)(-\sqrt{3}) = -2\sqrt{3} \)
- \( 2(\sqrt{2})(-\sqrt{3}) = -2\sqrt{6} \)

Putting it all together:

\[ b^2 = 1 + 2 + 3 + 2\sqrt{2} - 2\sqrt{3} - 2\sqrt{6} = 6 + 2\sqrt{2} - 2\sqrt{3} - 2\sqrt{6} \]

### Step 3: Calculate \( a^2 + b^2 \)

Now we add \( a^2 \) and \( b^2 \):

\[ a^2 + b^2 = (6 + 2\sqrt{2} + 2\sqrt{3} + 2\sqrt{6}) + (6 + 2\sqrt{2} - 2\sqrt{3} - 2\sqrt{6}) \]

Combining like terms:

\[ = 6 + 6 + 2\sqrt{2} + 2\sqrt{2} + 2\sqrt{3} - 2\sqrt{3} + 2\sqrt{6} - 2\sqrt{6} \]

This simplifies to:

\[ = 12 + 4\sqrt{2} \]

### Step 4: Calculate \( -2a - 2b \)

Now we calculate \( -2a - 2b \):

\[ -2a = -2(1 + \sqrt{2} + \sqrt{3}) = -2 - 2\sqrt{2} - 2\sqrt{3} \]

\[ -2b = -2(1 + \sqrt{2} - \sqrt{3}) = -2 - 2\sqrt{2} + 2\sqrt{3} \]

Adding these:

\[ -2a - 2b = (-2 - 2\sqrt{2} - 2\sqrt{3}) + (-2 - 2\sqrt{2} + 2\sqrt{3}) = -4 - 4\sqrt{2} \]

### Step 5: Combine Results

Now we combine the results from Steps 3 and 4:

\[ a^2 + b^2 - 2a - 2b = (12 + 4\sqrt{2}) + (-4 - 4\sqrt{2}) \]

This simplifies to:

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

### Final Answer

Thus, the final answer is:

\[ \boxed{8} \]