Let's simplify the given expression step by step. The expression is:

$\frac{2 + \sqrt{3}}{1 + \sqrt{2}}$

To simplify this, we'll multiply the numerator and denominator by the conjugate of the denominator to eliminate the square root from the denominator.

Step 1: Multiply by the conjugate

The conjugate of $1 + \sqrt{2}$ is $1 - \sqrt{2}$, so we multiply both the numerator and denominator by $1 - \sqrt{2}$:

$\frac{2 + \sqrt{3}}{1 + \sqrt{2}} \times \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{(2 + \sqrt{3})(1 - \sqrt{2})}{(1 + \sqrt{2})(1 - \sqrt{2})}$

Step 2: Simplify the denominator

The denominator is a difference of squares:

$(1 + \sqrt{2})(1 - \sqrt{2}) = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1$

Step 3: Expand the numerator

Now expand the numerator:

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

So, the expression becomes:

$\frac{2 - 2\sqrt{2} + \sqrt{3} - \sqrt{6}}{-1}$

Step 4: Simplify the expression

We can simplify by distributing the negative sign in the denominator:

$-(2 - 2\sqrt{2} + \sqrt{3} - \sqrt{6}) = -2 + 2\sqrt{2} - \sqrt{3} + \sqrt{6}$

Thus, the simplified expression is:

$2\sqrt{2} + \sqrt{6} - \sqrt{3} - 2$

Step 5: Compare with the options

The correct choice is the second option:

$2\sqrt{2} + \sqrt{6} - \sqrt{3} - 2$

Would you like further details on this solution, or have any questions?

Here are 5 related questions:

1. How do you find the conjugate of an expression involving square roots?
2. What is the difference of squares formula, and why is it useful?
3. Can the conjugate method be used for any radical expression?
4. How do you rationalize a denominator with two terms (binomial)?
5. How do you simplify an expression involving square roots of different numbers?

Tip: Multiplying by the conjugate is a common method to simplify expressions with square roots in the denominator.