The expression you provided is:

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

We will simplify this expression. To begin, we can rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator, $4 - 4\sqrt{2}$, to eliminate the square root from the denominator.

Step 1: Multiply numerator and denominator by the conjugate of the denominator.

$\frac{-2 + \sqrt{5}}{4 + 4\sqrt{2}} \times \frac{4 - 4\sqrt{2}}{4 - 4\sqrt{2}} = \frac{(-2 + \sqrt{5})(4 - 4\sqrt{2})}{(4 + 4\sqrt{2})(4 - 4\sqrt{2})}$

Step 2: Simplify the denominator.

The denominator is a difference of squares: $(4 + 4\sqrt{2})(4 - 4\sqrt{2}) = 4^2 - (4\sqrt{2})^2 = 16 - 16 \times 2 = 16 - 32 = -16$

Step 3: Expand the numerator.

We need to multiply $(-2 + \sqrt{5})$ by $(4 - 4\sqrt{2})$:

$(-2 + \sqrt{5})(4 - 4\sqrt{2}) = (-2)(4) + (-2)(-4\sqrt{2}) + (\sqrt{5})(4) + (\sqrt{5})(-4\sqrt{2})$ $= -8 + 8\sqrt{2} + 4\sqrt{5} - 4\sqrt{10}$ Thus, the numerator is: $-8 + 8\sqrt{2} + 4\sqrt{5} - 4\sqrt{10}$

Step 4: Put it all together.

Now the expression becomes: $\frac{-8 + 8\sqrt{2} + 4\sqrt{5} - 4\sqrt{10}}{-16}$

Step 5: Simplify the fraction.

We can now simplify each term by dividing by $-16$:

$\frac{-8}{-16} + \frac{8\sqrt{2}}{-16} + \frac{4\sqrt{5}}{-16} + \frac{-4\sqrt{10}}{-16}$ $= \frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{5}}{4} + \frac{\sqrt{10}}{4}$

Thus, the simplified expression is: $\frac{1}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{5}}{4} + \frac{\sqrt{10}}{4}$

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Would you like any further details or clarifications on this solution?

Here are some additional questions related to the topic:

1. What is rationalization and why do we use it in simplifying expressions with square roots?
2. How do we multiply square roots in algebraic expressions?
3. What are some other methods to simplify algebraic fractions?
4. How can you check if an expression is fully simplified?
5. Can we rationalize a denominator with multiple terms? If so, how?

Tip: Always remember that when rationalizing denominators, you should multiply by the conjugate if the denominator involves a binomial expression with square roots.