The expression in the image can be written as:

$\frac{(3 + \sqrt{8}) - \left(\frac{1}{3 + \sqrt{8}}\right)}{2}$

Step-by-step solution:

1. Simplify the expression: First, we need to simplify the numerator:

   $(3 + \sqrt{8}) - \frac{1}{3 + \sqrt{8}}$

2. Find a common denominator: The common denominator for this expression is $3 + \sqrt{8}$, so we rewrite the first term as:

   $\frac{(3 + \sqrt{8})(3 + \sqrt{8})}{3 + \sqrt{8}} - \frac{1}{3 + \sqrt{8}}$

3. Multiply the first term: Now expand the multiplication $(3 + \sqrt{8})^2$:

   $(3 + \sqrt{8})^2 = 3^2 + 2 \cdot 3 \cdot \sqrt{8} + (\sqrt{8})^2 = 9 + 6\sqrt{8} + 8 = 17 + 6\sqrt{8}$

   So the expression becomes:

   $\frac{(17 + 6\sqrt{8}) - 1}{3 + \sqrt{8}} = \frac{16 + 6\sqrt{8}}{3 + \sqrt{8}}$

4. Simplify the overall fraction: The simplified numerator is $16 + 6\sqrt{8}$, and now we divide the entire expression by 2:

   $\frac{16 + 6\sqrt{8}}{2(3 + \sqrt{8})}$

   This is the final simplified expression.

Would you like more details on any step, or to proceed with numerical evaluation? Here are a few related questions you might find interesting:

1. How can this expression be approximated to a decimal form?
2. What would happen if $\sqrt{8}$ were replaced with another irrational number like $\sqrt{2}$?
3. Can this form be simplified further if we rationalize the denominator?
4. How does multiplying the numerator and denominator by the conjugate affect the result?
5. Could this form arise in a physical or geometric context, such as in coordinate geometry?

Tip: When simplifying expressions involving square roots, it can help to recognize patterns in binomial squares and common factorization techniques.