Let's solve the equations one by one to determine which is correct.

First equation:

$-5a^4(2a^2 + 4) = 10a^4 - 20a^2$ Expand the left side: $-5a^4(2a^2 + 4) = -5a^4 \cdot 2a^2 - 5a^4 \cdot 4 = -10a^6 - 20a^4$ The right side is $10a^4 - 20a^2$, which is clearly different. This equation is incorrect.

Second equation:

$-4x^2(2x^2 + 5) = -8x^4 - 20x^2$ Expand the left side: $-4x^2(2x^2 + 5) = -4x^2 \cdot 2x^2 - 4x^2 \cdot 5 = -8x^4 - 20x^2$ The right side is also $-8x^4 - 20x^2$, which is correct.

Third equation:

$-6y^4(4y^2 + 2) = -24y^8 - 12y^5$ Expand the left side: $-6y^4(4y^2 + 2) = -6y^4 \cdot 4y^2 - 6y^4 \cdot 2 = -24y^6 - 12y^4$ The right side is $-24y^8 - 12y^5$, which is different. This equation is incorrect.

Fourth equation:

$-4b^3(5b^2 + 3) = -20b^6 - 12b^4$ Expand the left side: $-4b^3(5b^2 + 3) = -4b^3 \cdot 5b^2 - 4b^3 \cdot 3 = -20b^5 - 12b^3$ The right side is $-20b^6 - 12b^4$, which is different. This equation is incorrect.

Conclusion:

The correct equation is the second one: $-4x^2(2x^2 + 5) = -8x^4 - 20x^2$

Would you like further details on the steps or explanations?

Here are some follow-up questions you might consider:

1. How do you distribute terms in polynomial equations?
2. Why is it important to expand both sides of an equation?
3. What happens when like terms don't match on both sides of an equation?
4. How can you simplify expressions involving exponents?
5. Can you solve similar problems with different coefficients or variables?

Tip: Always expand and simplify both sides of an equation before comparing them to check for correctness.