Let's begin by expanding and simplifying the given expression:

$-5(x - 4)(3x + 2)$

Step 1: Distribute $-5$ to the first set of parentheses $(x - 4)$ and then multiply by $(3x + 2)$.

We’ll first expand the product of $(x - 4)$ and $(3x + 2)$, then distribute the $-5$.

Step 2: Expand $(x - 4)(3x + 2)$.

$(x - 4)(3x + 2) = x(3x + 2) - 4(3x + 2)$ $= 3x^2 + 2x - 12x - 8$ $= 3x^2 - 10x - 8$

Step 3: Distribute $-5$ to the expanded expression.

$-5(3x^2 - 10x - 8) = -5 \cdot 3x^2 + (-5) \cdot (-10x) + (-5) \cdot (-8)$ $= -15x^2 + 50x + 40$

So, the expanded and simplified expression is:

$-15x^2 + 50x + 40$

If you need any clarification or more details, feel free to ask!

Here are some follow-up questions to expand your understanding:

1. How can we factor this expanded expression back?
2. What would happen if we changed the coefficient of $-5$?
3. How do we approach more complex expressions with multiple terms?
4. How do we combine like terms when simplifying expressions?
5. How would this expansion change if we had more terms in the binomials?

Tip: Always distribute each term in the parentheses carefully and double-check your signs to avoid small errors!