I'll solve the problems in the image step by step:

1. Simplify the following expression: $3x + 2y - 5x + 4y$

Combine like terms: $(3x - 5x) + (2y + 4y) = -2x + 6y$ Answer: $-2x + 6y$

2. Use the distributive property to expand: $4(x - 3)$

Apply the distributive property: $4(x - 3) = 4x - 12$ Answer: $4x - 12$

3. Factor the following expression: $12a + 3a$

Find the greatest common factor (GCF), which is $3a$: $12a + 3a = 3a(4 + 1)$ Answer: $3a(4 + 1)$ or $3a(5)$

4. Simplify: $2(x + 5) - 3(x - 1)$

Expand each term: $2(x + 5) = 2x + 10, \quad -3(x - 1) = -3x + 3$ Combine the expanded terms: $2x + 10 - 3x + 3 = (2x - 3x) + (10 + 3) = -x + 13$ Answer: $-x + 13$

5. Distribute and simplify: $5(2x - 1) + 2(x + 3)$

Expand each term: $5(2x - 1) = 10x - 5, \quad 2(x + 3) = 2x + 6$ Combine the expanded terms: $10x - 5 + 2x + 6 = (10x + 2x) + (-5 + 6) = 12x + 1$ Answer: $12x + 1$

6. Factor out the greatest common factor: $15x + 25x$

The GCF is $5x$: $15x + 25x = 5x(3 + 5)$ Answer: $5x(3 + 5)$ or $5x(8)$

7. Simplify the expression: $4y - (3y + 2) + (y - 5)$

Distribute and simplify: $4y - (3y + 2) + (y - 5) = 4y - 3y - 2 + y - 5$ Combine like terms: $(4y - 3y + y) + (-2 - 5) = 2y - 7$ Answer: $2y - 7$

Let me know if you'd like further details or have questions! 😊

Related Questions:

1. How do you identify and combine like terms in algebraic expressions?
2. What is the distributive property, and why is it useful?
3. How do you determine the greatest common factor of algebraic terms?
4. How do parentheses affect operations in algebraic expressions?
5. What are some strategies for simplifying complex expressions efficiently?

Tip:

When factoring, always check for the highest common factor to ensure maximum simplification!