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Let's solve the given expression step-by-step. The expression is:

\[
\frac{10a^2 - 5a^4}{20 - 10a^2} \cdot \frac{5}{a^2}
\]

### Step 1: Simplify the first fraction

The numerator is \( 10a^2 - 5a^4 \). We can factor out the common term:

\[
10a^2 - 5a^4 = 5a^2(2 - a^2)
\]

The denominator is \( 20 - 10a^2 \). We can factor out the common term:

\[
20 - 10a^2 = 10(2 - a^2)
\]

Now, rewrite the first fraction:

\[
\frac{10a^2 - 5a^4}{20 - 10a^2} = \frac{5a^2(2 - a^2)}{10(2 - a^2)}
\]

### Step 2: Cancel out the common terms

Since \( 2 - a^2 \) appears in both the numerator and denominator, and it is not zero, we can cancel it out:

\[
\frac{5a^2(2 - a^2)}{10(2 - a^2)} = \frac{5a^2}{10} = \frac{a^2}{2}
\]

### Step 3: Combine with the second fraction

Now we multiply this simplified expression by the second fraction:

\[
\frac{a^2}{2} \cdot \frac{5}{a^2}
\]

Cancel out \( a^2 \) from both the numerator and denominator:

\[
= \frac{1}{2} \cdot 5 = \frac{5}{2}
\]

### Final Answer:
The simplified result of the given expression is:

\[
\boxed{\frac{5}{2}}
\]

Would you like more details or further clarification?

### Related Questions:
1. How do you factor polynomials like \( 10a^2 - 5a^4 \)?
2. How do you simplify expressions with variables in both the numerator and denominator?
3. What are the conditions for cancelling terms in algebraic fractions?
4. How can factoring help in simplifying complex algebraic expressions?
5. What happens if the term \( 2 - a^2 \) equals zero? How does it affect the solution?

### Tip:
Always check for common factors in both the numerator and denominator early on. This helps in simplifying expressions more efficiently!

Simplify the expression: \( \frac{10a^2 - 5a^4}{20 - 10a^2} \cdot \frac{5}{a^2} \)

Learn how to simplify the expression \( \frac{10a^2 - 5a^4}{20 - 10a^2} \cdot \frac{5}{a^2} \) by factoring polynomials and canceling terms. This example is ideal for students in Grades 9-11, focusing on key algebraic concepts such as rational expressions, factorization, and simplification.

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