$$ 2\sqrt{5\left(3\sqrt{10-4\sqrt{5)}}\right)} $$

Let's simplify the expression $ 2\sqrt{5\left(3\sqrt{10-4\sqrt{5)}}\right)} $ step by step.

### Step 1: Simplify the expression inside the square root

Start by simplifying the innermost expression:
\[
\sqrt{10 - 4\sqrt{5}}
\]

We can try expressing this as:
\[
\sqrt{a} - \sqrt{b}
\]
where $ a $ and $ b $ are positive numbers.

Square both sides:
\[
10 - 4\sqrt{5} = a + b - 2\sqrt{ab}
\]

Comparing both sides, we get two equations:
1. $ a + b = 10 $
2. $ -2\sqrt{ab} = -4\sqrt{5} $

From the second equation:
\[
\sqrt{ab} = 2\sqrt{5}
\]
Square both sides:
\[
ab = 20
\]

Now, solve the system:
1. $ a + b = 10 $
2. $ ab = 20 $

These are the roots of the quadratic equation:
\[
t^2 - 10t + 20 = 0
\]

Solving this quadratic equation:
\[
t = \frac{10 \pm \sqrt{100 - 80}}{2} = \frac{10 \pm \sqrt{20}}{2} = \frac{10 \pm 2\sqrt{5}}{2}
\]
\[
t = 5 \pm \sqrt{5}
\]

So, $ a = 5 + \sqrt{5} $ and $ b = 5 - \sqrt{5} $.

Thus:
\[
\sqrt{10 - 4\sqrt{5}} = \sqrt{5+\sqrt{5}} - \sqrt{5-\sqrt{5}}
\]

### Step 2: Substitute and simplify the expression
Substitute this into the original expression:
\[
2\sqrt{5\left(3\sqrt{5+\sqrt{5}} - \sqrt{5-\sqrt{5}}\right)}
\]

This expression can get quite complicated without specific substitutions or identities, but typically the strategy would involve:
- Trying to simplify each radical,
- Looking for patterns or symmetry, 
- Applying approximations if needed.

However, it's possible that further simplifications might only be numerical unless a specific identity can be applied. 

Do you want to delve further into each step or explore approximations?

---
### Related Questions:
1. How would you solve $ \sqrt{10 - 4\sqrt{5}} $ by completing the square?
2. What techniques can simplify nested radicals effectively?
3. How do you solve the quadratic equation $ t^2 - 10t + 20 = 0 $?
4. How does the quadratic formula work, and how does it relate to this problem?
5. What are some common strategies for dealing with expressions under square roots in general?

### Tip:
For nested radicals, it's often helpful to first test if they can be expressed as a difference or sum of two square roots. This can make the overall simplification process much easier.

Explore how to simplify the expression 2 sqrt(5(3 sqrt(10-4 sqrt(5)))) step-by-step. This problem involves advanced techniques in nested radicals and quadratic equations.

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