Let's go step by step to simplify each expression.

Part 1: Simplifying expressions

1. $-4x + 5x$ $= (5x - 4x) = x$

2. $4n + 4 + 1 + 3n$ $= (4n + 3n) + (4 + 1) = 7n + 5$

3. $-2x - 8 - 7x + 2$ $= (-2x - 7x) + (-8 + 2) = -9x - 6$

4. $-8x - 10x$ $= (-8x - 10x) = -18x$

5. $2k - k$ $= (2k - k) = k$

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Part 2: Using the distributive property to simplify each expression

1. $3(6n - 8)$ $= 3(6n) - 3(8) = 18n - 24$

2. $8(r + 1)$ $= 8r + 8(1) = 8r + 8$

3. $-5(8v - 2)$ $= -5(8v) + (-5)(-2) = -40v + 10$

4. $-(3a - 3)$ $= -1(3a - 3) = -3a + 3$

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Part 3: Using the distributive property, then combining like terms

1. $-n + 4(n + 1)$ $= -n + 4(n) + 4(1) = -n + 4n + 4 = 3n + 4$

2. $2x - 3(3x - 1)$ $= 2x - 3(3x) + 3(1) = 2x - 9x + 3 = -7x + 3$

3. $7 - 2(-3k + 4)$ $= 7 - 2(-3k) - 2(4) = 7 + 6k - 8 = 6k - 1$

4. $-3p - (-8 + 4p)$ $= -3p + 8 - 4p = (-3p - 4p) + 8 = -7p + 8$

5. $-4 + 6(-4x + 3)$ $= -4 + 6(-4x) + 6(3) = -4 - 24x + 18 = -24x + 14$

6. $3n + 3(1 + 8n)$ $= 3n + 3(1) + 3(8n) = 3n + 3 + 24n = 27n + 3$

7. $-2 - 5(4 + 3r)$ $= -2 - 5(4) - 5(3r) = -2 - 20 - 15r = -22 - 15r$

8. $-1 - 3(m - 4)$ $= -1 - 3(m) + 3(4) = -1 - 3m + 12 = -3m + 11$

9. $-2n - (n + 2)$ $= -2n - 1(n + 2) = -2n - n - 2 = -3n - 2$

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Would you like further explanations on any of the steps or more details?

Five Related Questions:

1. How do you identify like terms when simplifying algebraic expressions?
2. How does the distributive property work when there are multiple terms inside parentheses?
3. Can you apply the distributive property in reverse to factor expressions?
4. What’s the difference between simplifying and solving an equation?
5. How can combining like terms help in solving equations?

Tip: When applying the distributive property, ensure you distribute the sign (positive or negative) to each term inside the parentheses.