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To simplify the expression \((4x + 1)(3x + 5)\), we can apply the distributive property (also known as the FOIL method for binomials).

### Step-by-Step Expansion:

1. **First**: Multiply the first terms in each binomial:
   \[
   4x \cdot 3x = 12x^2
   \]

2. **Outside**: Multiply the outer terms:
   \[
   4x \cdot 5 = 20x
   \]

3. **Inside**: Multiply the inner terms:
   \[
   1 \cdot 3x = 3x
   \]

4. **Last**: Multiply the last terms:
   \[
   1 \cdot 5 = 5
   \]

### Combine Like Terms:
Now, combine the terms:
\[
12x^2 + 20x + 3x + 5
\]
\[
12x^2 + (20x + 3x) + 5 = 12x^2 + 23x + 5
\]

So, the simplified answer is:
\[
12x^2 + 23x + 5
\]

Would you like further details on any of these steps? 

Here are 5 related questions you can explore:
1. What is the distributive property, and how is it used in binomial expansion?
2. How does the FOIL method work for multiplying binomials?
3. Can you explain the process of combining like terms in detail?
4. What happens when you multiply two binomials that include negative terms?
5. How can you apply this method to more complex polynomials?

**Tip:** When expanding binomials, always ensure to multiply each term in the first binomial by each term in the second one.

Learn how to simplify the expression (4x + 1)(3x + 5) step by step using the FOIL method. This problem covers key algebraic concepts such as the distributive property and combining like terms, suitable for students in Grades 8-10.

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