Use the FOIL method to multiply the binomials.

To multiply the binomials \( (5x^3 + 6)(x^2 - 4) \) using the FOIL method, we'll follow the steps for each part of the method: First, Outside, Inside, Last.

1. **First**: Multiply the first terms.
\[
5x^3 \cdot x^2 = 5x^{3+2} = 5x^5
\]

2. **Outside**: Multiply the outer terms.
\[
5x^3 \cdot (-4) = -20x^3
\]

3. **Inside**: Multiply the inner terms.
\[
6 \cdot x^2 = 6x^2
\]

4. **Last**: Multiply the last terms.
\[
6 \cdot (-4) = -24
\]

Now, combine all these results together:
\[
5x^5 - 20x^3 + 6x^2 - 24
\]

Next, we will rearrange the terms in descending order of the exponent:
\[
5x^5 - 20x^3 + 6x^2 - 24
\]

Thus, the final answer after simplifying is:
\[
\boxed{5x^5 - 20x^3 + 6x^2 - 24}
\]