The image contains a list of algebraic expressions and equations. I'll start by writing them down to help analyze and solve any specific ones you may be interested in. Here's what I see from the image:

1. $8x^7 - 7x^7y + 10x^6y - 12xy^6$
2. $(x + 1)^7 - (x + 5)^7$
3. $x^7 + 7x + 1$
4. $x^7 + 7x + 12$
5. (empty space, likely missing expression)
6. $y^7 - 6y - 12$
7. $x^7 - y^5$
8. $6x^7 - (y - 7y)^7$
9. $x^7 + 1$
10. $6x^7 - (x - 7)^7$
11. $x^7 - 9$

Let me know if you'd like to solve or simplify any of these expressions, or provide further details! Here are a few questions that might guide further inquiry:

1. Would you like to simplify or solve any of these equations?
2. Are you looking for solutions over real numbers, integers, or another domain?
3. Do you want help with any specific algebraic method, such as factoring or expanding?
4. Would you like to see step-by-step solutions for the more complex expressions?
5. Do any of these equations represent real-world problems you'd like assistance with?

Tip: When dealing with polynomials, it's often helpful to check if factoring or grouping terms can simplify the equation first.