If `a=2+sqrt(3)+sqrt(5)` and `b=3+sqrt(3)-sqrt(5)`, then `a^(2)+b^(2)-4a-6b-3` is equal to

Options

A. `2`
B. `-1`
C. `1`
D. `0`

Answer

To solve the expression \( a^2 + b^2 - 4a - 6b - 3 \) given \( a = 2 + \sqrt{3} + \sqrt{5} \) and \( b = 3 + \sqrt{3} - \sqrt{5} \), we will follow these steps:

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

Using the formula \( (x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + xz + yz) \), we can expand \( a^2 \).

Let:
- \( x = 2 \)
- \( y = \sqrt{3} \)
- \( z = \sqrt{5} \)

Then:
\[ a^2 = (2 + \sqrt{3} + \sqrt{5})^2 = 2^2 + (\sqrt{3})^2 + (\sqrt{5})^2 + 2(2\sqrt{3} + 2\sqrt{5} + \sqrt{3}\sqrt{5}) \]
Calculating each term:
\[ = 4 + 3 + 5 + 2(2\sqrt{3} + 2\sqrt{5} + \sqrt{15}) \]
\[ = 12 + 4\sqrt{3} + 4\sqrt{5} + 2\sqrt{15} \]

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

Using the same expansion method for \( b \):
\[ b^2 = (3 + \sqrt{3} - \sqrt{5})^2 = 3^2 + (\sqrt{3})^2 + (-\sqrt{5})^2 + 2(3\sqrt{3} - 3\sqrt{5} + \sqrt{3}(-\sqrt{5})) \]
Calculating each term:
\[ = 9 + 3 + 5 + 2(3\sqrt{3} - 3\sqrt{5} - \sqrt{15}) \]
\[ = 17 + 6\sqrt{3} - 6\sqrt{5} - 2\sqrt{15} \]

### Step 3: Combine \( a^2 \) and \( b^2 \)

Now we add \( a^2 \) and \( b^2 \):
\[ a^2 + b^2 = (12 + 4\sqrt{3} + 4\sqrt{5} + 2\sqrt{15}) + (17 + 6\sqrt{3} - 6\sqrt{5} - 2\sqrt{15}) \]
Combining like terms:
\[ = 29 + (4\sqrt{3} + 6\sqrt{3}) + (4\sqrt{5} - 6\sqrt{5}) + (2\sqrt{15} - 2\sqrt{15}) \]
\[ = 29 + 10\sqrt{3} - 2\sqrt{5} \]

### Step 4: Calculate \( 4a \) and \( 6b \)

Now we calculate \( 4a \) and \( 6b \):
\[ 4a = 4(2 + \sqrt{3} + \sqrt{5}) = 8 + 4\sqrt{3} + 4\sqrt{5} \]
\[ 6b = 6(3 + \sqrt{3} - \sqrt{5}) = 18 + 6\sqrt{3} - 6\sqrt{5} \]

### Step 5: Combine \( -4a - 6b \)

Now we compute \( -4a - 6b \):
\[ -4a - 6b = -(8 + 4\sqrt{3} + 4\sqrt{5}) - (18 + 6\sqrt{3} - 6\sqrt{5}) \]
\[ = -8 - 4\sqrt{3} - 4\sqrt{5} - 18 - 6\sqrt{3} + 6\sqrt{5} \]
Combining like terms:
\[ = -26 - (4\sqrt{3} + 6\sqrt{3}) + (6\sqrt{5} - 4\sqrt{5}) \]
\[ = -26 - 10\sqrt{3} + 2\sqrt{5} \]

### Step 6: Combine everything

Now we combine everything to find \( a^2 + b^2 - 4a - 6b - 3 \):
\[ = (29 + 10\sqrt{3} - 2\sqrt{5}) + (-26 - 10\sqrt{3} + 2\sqrt{5}) - 3 \]
\[ = 29 - 26 - 3 + (10\sqrt{3} - 10\sqrt{3}) + (-2\sqrt{5} + 2\sqrt{5}) \]
\[ = 0 \]

### Final Answer
Thus, the value of \( a^2 + b^2 - 4a - 6b - 3 \) is \( \boxed{0} \).