Certainly! Let's apply the distributive property to each of the expressions in detail.

Detailed Step-by-Step Solutions Using the Distributive Property

1. Expression: $-46 \times 999$

   Using the distributive property: $-46 \times 999 = -46 \times (1000 - 1)$ Distribute $-46$: $= (-46 \times 1000) + (-46 \times -1) = -46000 + 46 = -45954$

2. Expression: $-56 \times 35 + (-56) \times 15$

   Notice that $-56$ is a common factor: $-56 \times 35 + (-56) \times 15 = -56(35 + 15)$ Add the numbers inside the parentheses: $= -56 \times 50 = -2800$

3. Expression: $-63 \times -99$

   Using the distributive property: $-63 \times -99 = (-63) \times (-(100 - 1)) = -63 \times -100 + 63 \times 1$ Simplify each term: $= 6300 + 63 = 6237$

4. Expression: $58 \times (-17) + (-58) \times 3$

   Notice that $58$ is a common factor with opposite signs: $58 \times (-17) + (-58) \times 3 = 58 \times (-17) - 58 \times 3$ Factor out the common term: $= 58(-17 - 3) = 58 \times -20 = -1160$

Final Distributed Property Solutions

1. $-46 \times 999 = -45954$
2. $-56 \times 35 + (-56) \times 15 = -2800$
3. $-63 \times -99 = 6237$
4. $58 \times (-17) + (-58) \times 3 = -1160$

Would you like more details or have any questions? Here are five more questions you might be interested in:

1. How do you solve linear equations?
2. What are the properties of exponents?
3. How do you find the derivative of a function?
4. What is the quadratic formula and how is it used?
5. How do you perform matrix multiplication?

Tip: Remember that the distributive property states $a(b + c) = ab + ac$. This can simplify calculations and is particularly useful for mental math.