Let's go through each problem and simplify the expressions one by one.

8. $3x(x - 2) + 7(x - 2)$

Let's factor out the common term $(x - 2)$: $3x(x - 2) + 7(x - 2) = (3x + 7)(x - 2)$

9. $4x(3x + 5) - 3(3x + 5)$

Factor out the common term $(3x + 5)$: $4x(3x + 5) - 3(3x + 5) = (4x - 3)(3x + 5)$

10. $x(4x + 9) + (4x + 9)$

Factor out the common term $(4x + 9)$: $x(4x + 9) + (4x + 9) = (x + 1)(4x + 9)$

11. $8a - 2ab + 6ac - 4a^2$

Group and factor terms: $8a - 2ab + 6ac - 4a^2 = 2a(4 - b + 3c - 2a)$

12. $15x^2 - 20x + 10$

Factor out the greatest common factor, which is 5: $15x^2 - 20x + 10 = 5(3x^2 - 4x + 2)$

If you would like more detailed explanations on how each of these were simplified or factored, let me know!

Relative Questions to Expand Understanding:

1. How do you identify a common factor in algebraic expressions?
2. What is the process of factoring a polynomial?
3. How can you use the distributive property to simplify expressions?
4. What are some strategies for factoring trinomials?
5. How does factoring help in solving equations?

Tip:

When factoring, always look for a common factor first before attempting more complex methods like grouping or using special formulas.