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!

Learn how to simplify the radical expression 2(2√10 + 2√8) - √2 / (√5 - 2) using key concepts like radical simplification and rationalizing the denominator. This step-by-step solution explains core formulas and methods. Suitable for Grades 9-11.

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