Let's simplify each expression using the distributive property.

The distributive property states that $a(b + c) = ab + ac$.

a. $8(x + 4)$

Apply the distributive property: $8(x + 4) = 8 \cdot x + 8 \cdot 4 = 8x + 32$ So, the simplified expression is $8x + 32$.

b. $x^2(3 - x)$

Apply the distributive property: $x^2(3 - x) = x^2 \cdot 3 - x^2 \cdot x = 3x^2 - x^3$ So, the simplified expression is $3x^2 - x^3$.

c. $5x(x + 3)$

Apply the distributive property: $5x(x + 3) = 5x \cdot x + 5x \cdot 3 = 5x^2 + 15x$ So, the simplified expression is $5x^2 + 15x$.

d. $4x^2(8x - 5)$

Apply the distributive property: $4x^2(8x - 5) = 4x^2 \cdot 8x - 4x^2 \cdot 5 = 32x^3 - 20x^2$ So, the simplified expression is $32x^3 - 20x^2$.

If you have any further questions or need more details, feel free to ask!

Here are 5 related questions you might want to consider:

1. What is the distributive property and why is it useful in algebra?
2. How would the expressions change if we had subtraction instead of addition inside the parentheses?
3. Can you apply the distributive property when there are more than two terms inside the parentheses?
4. How can you factor expressions to reverse the distributive property?
5. What happens when the distributive property involves more complex expressions, such as binomials?

Tip: Always double-check your multiplication and signs when using the distributive property to avoid simple mistakes.