The given expression is:

$\frac{2\sqrt{2} - 2\sqrt{5}}{4\sqrt{3} + 4\sqrt{2}}.$

To simplify it by removing square roots from the denominator, we rationalize the denominator. Here's how:

Step 1: Factorize common terms in the numerator and denominator

Factor $2$ from the numerator and $4$ from the denominator:

$\frac{2(\sqrt{2} - \sqrt{5})}{4(\sqrt{3} + \sqrt{2})}.$

Simplify further: $\frac{\sqrt{2} - \sqrt{5}}{2(\sqrt{3} + \sqrt{2})}.$

Step 2: Rationalize the denominator

To rationalize $2(\sqrt{3} + \sqrt{2})$, multiply numerator and denominator by the conjugate of the denominator: $2(\sqrt{3} - \sqrt{2})$.

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

Step 3: Expand the numerator

Expand $(\sqrt{2} - \sqrt{5})(\sqrt{3} - \sqrt{2})$ using the distributive property:

$(\sqrt{2} - \sqrt{5})(\sqrt{3} - \sqrt{2}) = \sqrt{6} - 2 - \sqrt{15} + \sqrt{10}.$

Thus, the numerator becomes: $\sqrt{6} - \sqrt{15} + \sqrt{10} - 2.$

Step 4: Simplify the denominator

The denominator is: $2(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}).$

Use the difference of squares formula: $(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2}) = 3 - 2 = 1.$

So the denominator simplifies to: $2 \cdot 1 = 2.$

Step 5: Combine

The simplified expression is: $\frac{\sqrt{6} - \sqrt{15} + \sqrt{10} - 2}{2}.$

You can also distribute the denominator: $\frac{\sqrt{6}}{2} - \frac{\sqrt{15}}{2} + \frac{\sqrt{10}}{2} - 1.$

Thus, the final simplified expression is: $\frac{\sqrt{6} - \sqrt{15} + \sqrt{10} - 2}{2}.$

Let me know if you'd like further clarification!

Related Questions:

1. How do you rationalize a denominator with more complex conjugates?
2. What is the difference of squares formula and why is it useful here?
3. What happens if the denominator is a cube root instead of a square root?
4. How can the distributive property be used to simplify the numerator faster?
5. How can we confirm our rationalization process is correct?

Tip: Always simplify by factoring common terms before rationalizing—it can make calculations cleaner and easier!