The given expression is:

$\frac{4(5^{2n-1}) - 10(5^{2n-1})}{2(5^{2n})}$

Let's simplify this step by step:

Step 1: Simplify the numerator.

The numerator is: $4(5^{2n-1}) - 10(5^{2n-1})$ Factor out $5^{2n-1}$ from both terms: $5^{2n-1}(4 - 10) = 5^{2n-1}(-6)$

Step 2: Rewrite the expression.

Now the expression becomes: $\frac{-6(5^{2n-1})}{2(5^{2n})}$

Step 3: Simplify the denominator.

We know that $5^{2n} = 5^{2n-1} \cdot 5$, so rewrite the denominator as: $2(5^{2n-1} \cdot 5)$

Step 4: Cancel out common factors.

The term $5^{2n-1}$ appears in both the numerator and denominator, so we can cancel it out: $\frac{-6}{2 \cdot 5}$

Step 5: Final simplification.

Simplify the remaining expression: $\frac{-6}{10} = -\frac{3}{5}$

Thus, the simplified expression is: $-\frac{3}{5}$

Do you need further details or have any questions?

Related questions:

1. How would the expression change if the power of 5 was different?
2. Can you generalize this method for other base powers?
3. What would happen if there were additional terms in the numerator?
4. How does factoring help in simplifying complex expressions like this one?
5. How would the solution change if the denominator had a different factor?

Tip: Always look for common factors to simplify both the numerator and denominator when possible.