Let's simplify each expression from the image. I'll solve these step by step:

1. $5(x + 2)$

Distribute $5$: $5x + 10$

2. $7x(x + y)$

Distribute $7x$: $7x^2 + 7xy$

3. $5x(x^2 + 10)$

Distribute $5x$: $5x^3 + 50x$

4. $10y(6xy - 14)$

Distribute $10y$: $60xy^2 - 140y$

5. $-2x^2(-2 + 6xy)$

Distribute $-2x^2$: $4x^2 - 12x^3y$

6. $-6x(8 - 6x^3)$

Distribute $-6x$: $-48x + 36x^4$

7. $(x + 2)(x + 5)$

Use the distributive property (FOIL method): $x^2 + 5x + 2x + 10 = x^2 + 7x + 10$

8. $(x - 6)(x + 9)$

Use the distributive property (FOIL method): $x^2 + 9x - 6x - 54 = x^2 + 3x - 54$

9. $(x + 3)(x + 3)$

This is a perfect square trinomial: $x^2 + 3x + 3x + 9 = x^2 + 6x + 9$

10. $(x - 5)(x + 5)$

This is a difference of squares: $x^2 - 25$

Do you want more details on any step? 😊 Here are some related questions to extend your learning:

1. How do you determine when to use FOIL or direct distribution?
2. What happens if the constants in the parentheses are fractions?
3. Can you apply these concepts to factorization?
4. Why does $(x - 5)(x + 5)$ simplify to a difference of squares?
5. How does simplifying change when additional variables are involved?

Tip: For polynomials, always check for common factors before distributing—this simplifies your work.