https://file.aiquickdraw.com/mathbot/file/1726429169034uluujzix.jpg

Let's solve the two equations by expanding the terms inside the parentheses and matching the results with the format \( ax^2 + bx + c \).

### First equation: 
\[
(2x + 3)(4x + 2)
\]

We will apply the distributive property (FOIL method):
- First: \(2x \cdot 4x = 8x^2\)
- Outside: \(2x \cdot 2 = 4x\)
- Inside: \(3 \cdot 4x = 12x\)
- Last: \(3 \cdot 2 = 6\)

Now, combine like terms:
\[
8x^2 + (4x + 12x) + 6 = 8x^2 + 16x + 6
\]

So, the first equation becomes:
\[
8x^2 + 16x + 6
\]

### Second equation: 
\[
(3x + 4)(2x - 5)
\]

Again, using the distributive property (FOIL method):
- First: \(3x \cdot 2x = 6x^2\)
- Outside: \(3x \cdot (-5) = -15x\)
- Inside: \(4 \cdot 2x = 8x\)
- Last: \(4 \cdot (-5) = -20\)

Combine like terms:
\[
6x^2 + (-15x + 8x) - 20 = 6x^2 - 7x - 20
\]

So, the second equation becomes:
\[
6x^2 - 7x - 20
\]

### Final Answers:
- For \( (2x + 3)(4x + 2) \), the result is \( 8x^2 + 16x + 6 \).
- For \( (3x + 4)(2x - 5) \), the result is \( 6x^2 - 7x - 20 \).

If you'd like more details or have further questions, feel free to ask!

### Follow-up questions:
1. How does the FOIL method work in other similar problems?
2. Can we use the distributive property for more than two terms inside parentheses?
3. What is the significance of combining like terms after applying FOIL?
4. How can factoring help to verify the expanded form?
5. What other methods exist for expanding binomial products?

### Tip:
Always ensure you carefully combine like terms to avoid errors in the final expression!

In this lesson, we solve two polynomial expressions using the distributive property and the FOIL method. We expand (2x + 3)(4x + 2) and (3x + 4)(2x - 5), step by step, to arrive at simplified quadratic forms. Suitable for Grades 7-9.

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