It looks like you have a problem involving radicals and their simplification. Here's a breakdown of what I see:

You have an expression involving square roots that needs to be simplified:

$\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{5}} \times \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} - \sqrt{5}} = a - b \sqrt{6}$

It seems you need to simplify the left-hand side and express it in the form $a - b\sqrt{6}$.

Step-by-Step Solution:

1. Combine the two fractions:

   The multiplication of two fractions can be simplified by multiplying the numerators and the denominators together. So,

   $\frac{(\sqrt{3} - \sqrt{2}) (\sqrt{3} + \sqrt{5})}{(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})}$

2. Simplify the denominator using the difference of squares:

   The denominator is of the form $(a + b)(a - b) = a^2 - b^2$, where $a = \sqrt{3}$ and $b = \sqrt{5}$.

   $(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2$

   So, the denominator becomes $-2$.

3. Expand the numerator:

   Now, expand the expression $(\sqrt{3} - \sqrt{2})(\sqrt{3} + \sqrt{5})$ using the distributive property:

   $= \sqrt{3}\cdot\sqrt{3} + \sqrt{3}\cdot\sqrt{5} - \sqrt{2}\cdot\sqrt{3} - \sqrt{2}\cdot\sqrt{5}$

   Simplifying each term:

   $= 3 + \sqrt{15} - \sqrt{6} - \sqrt{10}$

4. Put it all together:

   So, the expression becomes:

   $\frac{3 + \sqrt{15} - \sqrt{6} - \sqrt{10}}{-2}$

   Now, divide each term by $-2$:

   $= \frac{3}{-2} + \frac{\sqrt{15}}{-2} - \frac{\sqrt{6}}{-2} - \frac{\sqrt{10}}{-2}$

   Simplifying further:

   $= -\frac{3}{2} - \frac{\sqrt{15}}{2} + \frac{\sqrt{6}}{2} + \frac{\sqrt{10}}{2}$

5. Express it in the form $a - b \sqrt{6}$:

   Comparing this with $a - b\sqrt{6}$, you can see that:

   • $a = -\frac{3}{2} - \frac{\sqrt{15}}{2} + \frac{\sqrt{10}}{2}$
   • $b = -\frac{1}{2}$

Thus, the simplified form is:

$a = -\frac{3}{2} - \frac{\sqrt{15}}{2} + \frac{\sqrt{10}}{2}, \quad b = -\frac{1}{2}$

Let me know if you'd like more details on any step or further clarification!

Related Questions:

1. How do you simplify expressions involving radicals?
2. What are the properties of the difference of squares for square roots?
3. Can you explain how to rationalize denominators with complex square roots?
4. How do you expand binomials with square roots?
5. What is the general method to express radical expressions in a simplified form?

Tip: When simplifying expressions with square roots, remember that multiplying conjugates helps eliminate the square roots in the denominator!