Let's begin by solving the problems in the image you've uploaded.

1. $(x + 2)(x + 4)$

We apply the distributive property (FOIL method):

$(x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8$

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

Using the distributive property:

$(x - 5)(x + 6) = x^2 + 6x - 5x - 30 = x^2 + x - 30$

3. $(y + 6)(y - 10)$

Using the distributive property:

$(y + 6)(y - 10) = y^2 - 10y + 6y - 60 = y^2 - 4y - 60$

4. $(x - 8)(x - 7)$

Using the distributive property:

$(x - 8)(x - 7) = x^2 - 7x - 8x + 56 = x^2 - 15x + 56$

5. $(a + 2)(3a - 5)$

Using the distributive property:

$(a + 2)(3a - 5) = a(3a - 5) + 2(3a - 5) = 3a^2 - 5a + 6a - 10 = 3a^2 + a - 10$

6. $(z - 4)(6z + 1)$

Using the distributive property:

$(z - 4)(6z + 1) = z(6z + 1) - 4(6z + 1) = 6z^2 + z - 24z - 4 = 6z^2 - 23z - 4$

7. $(a + 3)(a^2 + 4a - 1)$

Using the distributive property:

$(a + 3)(a^2 + 4a - 1) = a(a^2 + 4a - 1) + 3(a^2 + 4a - 1) = a^3 + 4a^2 - a + 3a^2 + 12a - 3$ $= a^3 + 7a^2 + 11a - 3$

8. $(t - 2)(t^2 + 3t + 4)$

Using the distributive property:

$(t - 2)(t^2 + 3t + 4) = t(t^2 + 3t + 4) - 2(t^2 + 3t + 4)$ $= t^3 + 3t^2 + 4t - 2t^2 - 6t - 8 = t^3 + t^2 - 2t - 8$

9. $(4x - 2)(3x^2 - 2x - 5)$

Using the distributive property:

$(4x - 2)(3x^2 - 2x - 5) = 4x(3x^2 - 2x - 5) - 2(3x^2 - 2x - 5)$ $= 12x^3 - 8x^2 - 20x - 6x^2 + 4x + 10 = 12x^3 - 14x^2 - 16x + 10$

10. $(5y + 1)(2y^2 - y + 3)$

Using the distributive property:

$(5y + 1)(2y^2 - y + 3) = 5y(2y^2 - y + 3) + 1(2y^2 - y + 3)$ $= 10y^3 - 5y^2 + 15y + 2y^2 - y + 3 = 10y^3 - 3y^2 + 14y + 3$

Would you like any more details or clarification on any of these steps? Let me know!

Here are some follow-up questions for you:

1. How would you factor any of the expressions from these results?
2. Can you expand other binomials and polynomials?
3. How does the distributive property apply to binomials?
4. How do you use FOIL for binomials?
5. Can you identify any common patterns in these multiplication problems?

Tip: When multiplying polynomials, always remember to apply the distributive property (FOIL for binomials) carefully to ensure every term is multiplied.