Let's break this expression down and solve it step by step.

Given:
$2 \left( 2\sqrt{10} + 2\sqrt{8} \right) - \frac{\sqrt{2}}{\sqrt{5} - 2}$

Step 1: Simplify inside the parentheses

First, simplify the expression inside the parentheses: $2\left( 2\sqrt{10} + 2\sqrt{8} \right) = 2 \cdot 2\sqrt{10} + 2 \cdot 2\sqrt{8} = 4\sqrt{10} + 4\sqrt{8}$

Now, let's simplify $4\sqrt{8}$. Since $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$, we can write: $4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}$

So the expression now becomes: $4\sqrt{10} + 8\sqrt{2}$

Step 2: Deal with the fraction

Next, simplify the fraction $\frac{\sqrt{2}}{\sqrt{5} - 2}$.

To rationalize the denominator, multiply both the numerator and denominator by the conjugate of the denominator, $\sqrt{5} + 2$: $\frac{\sqrt{2}}{\sqrt{5} - 2} \cdot \frac{\sqrt{5} + 2}{\sqrt{5} + 2} = \frac{\sqrt{2}(\sqrt{5} + 2)}{(\sqrt{5})^2 - 2^2} = \frac{\sqrt{2}(\sqrt{5} + 2)}{5 - 4} = \frac{\sqrt{2}(\sqrt{5} + 2)}{1}$

Thus: $\frac{\sqrt{2}}{\sqrt{5} - 2} = \sqrt{2}(\sqrt{5} + 2)$

Step 3: Combine the two parts

Now, substitute back into the original expression: $4\sqrt{10} + 8\sqrt{2} - \sqrt{2}(\sqrt{5} + 2)$

Distribute $\sqrt{2}$ on the last term: $\sqrt{2}(\sqrt{5} + 2) = \sqrt{2} \cdot \sqrt{5} + \sqrt{2} \cdot 2 = \sqrt{10} + 2\sqrt{2}$

Now substitute this back into the expression: $4\sqrt{10} + 8\sqrt{2} - (\sqrt{10} + 2\sqrt{2})$

Step 4: Combine like terms

Now combine like terms: $(4\sqrt{10} - \sqrt{10}) + (8\sqrt{2} - 2\sqrt{2}) = 3\sqrt{10} + 6\sqrt{2}$

Final Answer:

$\boxed{3\sqrt{10} + 6\sqrt{2}}$

Would you like more details or have any other questions?

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Here are 5 related questions for further practice:

1. Simplify $\sqrt{18} + \sqrt{32}$.
2. Rationalize the denominator: $\frac{2\sqrt{3}}{\sqrt{7} - \sqrt{5}}$.
3. Simplify $3(4\sqrt{7} + \sqrt{3}) - 2\sqrt{3}$.
4. Simplify $2\left( \sqrt{12} + \sqrt{27} \right)$.
5. Rationalize the denominator: $\frac{\sqrt{6}}{\sqrt{2} + 1}$.

Tip: When rationalizing denominators, always look for the conjugate of the denominator to eliminate the square roots!