https://file.aiquickdraw.com/mathbot/file/17255305029672b0dtxds.jpg

The image didn't extract all the equations properly. However, I can see the overall structure of the problems from the image.

Let's manually address the visible equations:

1. \( (2a - 4b)(3a - 2b) \)
2. \( (100 - 1)(100 - 1) \)
3. \( \left( \frac{x}{y} - \frac{1}{5} \right) \left( \frac{x}{y} + \frac{1}{5} \right) \)
4. \( (7a^2 - 5b^2)^2 \)
5. \( (5m^2 - 7n)^3 \)

### Let's solve each of these step by step:

#### 1. \( (2a - 4b)(3a - 2b) \)
We expand this using distributive property (FOIL method):
\[
= (2a)(3a) + (2a)(-2b) + (-4b)(3a) + (-4b)(-2b)
= 6a^2 - 4ab - 12ab + 8b^2
\]
\[
= 6a^2 - 16ab + 8b^2
\]

#### 2. \( (100 - 1)(100 - 1) \)
This is a difference of squares:
\[
= (100)^2 - (1)^2 = 10000 - 1 = 9999
\]

#### 3. \( \left( \frac{x}{y} - \frac{1}{5} \right) \left( \frac{x}{y} + \frac{1}{5} \right) \)
This is also a difference of squares:
\[
= \left( \frac{x}{y} \right)^2 - \left( \frac{1}{5} \right)^2 
= \frac{x^2}{y^2} - \frac{1}{25}
\]

#### 4. \( (7a^2 - 5b^2)^2 \)
We expand this binomial:
\[
= (7a^2)^2 - 2(7a^2)(5b^2) + (5b^2)^2
= 49a^4 - 70a^2b^2 + 25b^4
\]

#### 5. \( (5m^2 - 7n)^3 \)
This is a binomial cube:
\[
= (5m^2)^3 - 3(5m^2)^2(7n) + 3(5m^2)(7n)^2 - (7n)^3
\]
\[
= 125m^6 - 3(25m^4)(7n) + 3(5m^2)(49n^2) - 343n^3
\]
\[
= 125m^6 - 525m^4n + 735m^2n^2 - 343n^3
\]

### Summary of Answers:
1. \( 6a^2 - 16ab + 8b^2 \)
2. \( 9999 \)
3. \( \frac{x^2}{y^2} - \frac{1}{25} \)
4. \( 49a^4 - 70a^2b^2 + 25b^4 \)
5. \( 125m^6 - 525m^4n + 735m^2n^2 - 343n^3 \)

Would you like more details or further clarification on any of these? 

Here are 5 related questions for further exploration:
1. How do you factor a quadratic expression?
2. What is the general formula for expanding binomials raised to powers?
3. Can you explain why the difference of squares method works?
4. How does the distributive property apply to polynomial multiplication?
5. How can we verify the solution to a binomial expansion?

**Tip:** When expanding polynomials, carefully track signs (positive/negative) to avoid mistakes in combining like terms.

Write the product of the following: 1) (2a - 4b)(3a - 2b) 2) (100 - 1)(100 - 1) 3) (x/y - 1/5)(x/y + 1/5) 4) (7a^2 - 5b^2)^2 5) (5m^2 - 7n)^3

Solve the polynomial and binomial multiplication problems, including expanding expressions like (2a - 4b)(3a - 2b) and (7a^2 - 5b^2)^2. This solution demonstrates the use of the distributive property, binomial expansion, and the difference of squares theorem.

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