Let's work through each part of the problem to simplify the expressions.

a) $5(x + 9) + 12x$

Distribute the $5$ to both terms inside the parentheses: $5(x + 9) = 5x + 45$ Now add the result to $12x$: $5x + 45 + 12x = 17x + 45$ Answer: $17x + 45$

b) $m + 3(2m - 4)$

Distribute the $3$ to both terms inside the parentheses: $3(2m - 4) = 6m - 12$ Now add $m$ to this result: $m + 6m - 12 = 7m - 12$ Answer: $7m - 12$

c) $x + 5 - 6(3x + 2)$

Distribute the $-6$ to both terms inside the parentheses: $-6(3x + 2) = -18x - 12$ Now combine like terms with the rest of the expression: $x + 5 - 18x - 12 = -17x - 7$ Answer: $-17x - 7$

d) $2(x + 7) + 3(x + 4)$

Distribute the $2$ and $3$ to the terms inside their respective parentheses: $2(x + 7) = 2x + 14 \quad \text{and} \quad 3(x + 4) = 3x + 12$ Now combine like terms: $2x + 14 + 3x + 12 = 5x + 26$ Answer: $5x + 26$

e) $-5(2x + 3) + (6x - 4)$

Distribute the $-5$ to both terms inside the parentheses: $-5(2x + 3) = -10x - 15$ Now combine like terms: $-10x - 15 + 6x - 4 = -4x - 19$ Answer: $-4x - 19$

f) $-6(2y + 5) + 3(5y - 1)$

Distribute the $-6$ and $3$ to the terms inside their respective parentheses: $-6(2y + 5) = -12y - 30 \quad \text{and} \quad 3(5y - 1) = 15y - 3$ Now combine like terms: $-12y - 30 + 15y - 3 = 3y - 33$ Answer: $3y - 33$

g) $(x - 4)(3) - (2x + 6)$

First, distribute the $3$ to both terms in the first set of parentheses: $(x - 4)(3) = 3x - 12$ Now subtract the entire second expression: $3x - 12 - (2x + 6) = 3x - 12 - 2x - 6$ Combine like terms: $(3x - 2x) + (-12 - 6) = x - 18$ Answer: $x - 18$

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

Distribute the $2$ and $-4$ to the terms inside the parentheses: $2(5x + 3) = 10x + 6 \quad \text{and} \quad -4(x + 3) = -4x - 12$ Now combine like terms: $10x + 6 - 4x - 12 = 6x - 6$ Answer: $6x - 6$

i) $-(-5m - 8) - 2(6m - 7) + 3(m - 1)$

First, distribute the $-1$ to $-(-5m - 8)$, which changes both signs: $-(-5m - 8) = 5m + 8$ Now distribute $-2$ and $3$ to their respective terms: $-2(6m - 7) = -12m + 14 \quad \text{and} \quad 3(m - 1) = 3m - 3$ Now combine like terms: $5m + 8 - 12m + 14 + 3m - 3 = (5m - 12m + 3m) + (8 + 14 - 3)$ $= -4m + 19$ Answer: $-4m + 19$

Would you like further details or explanations on any part of these solutions?

5 Related Questions:

1. How do you apply the distributive property in algebra?
2. How do you simplify expressions with multiple variables and terms?
3. What are some common mistakes to avoid when distributing negative signs?
4. How do you combine like terms after distribution?
5. How do you simplify complex expressions involving multiple parentheses?

Tip:

Always be careful with negative signs when distributing, as they can easily lead to mistakes in simplification.