To multiply the binomials \((4x + \frac{1}{2})(x + \frac{3}{2})\), we use the distributive property, often referred to as the FOIL method (First, Outer, Inner, Last):

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

2. **Outer:** Multiply the outer terms in the product:
   \[
   4x \cdot \frac{3}{2} = 6x
   \]

3. **Inner:** Multiply the inner terms:
   \[
   \frac{1}{2} \cdot x = \frac{1}{2}x
   \]

4. **Last:** Multiply the last terms:
   \[
   \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}
   \]

Now, combine all these results:
\[
4x^2 + 6x + \frac{1}{2}x + \frac{3}{4}
\]

Finally, combine like terms (the \(6x\) and \(\frac{1}{2}x\)):
\[
4x^2 + \frac{13}{2}x + \frac{3}{4}
\]

So, the expanded form of \((4x + \frac{1}{2})(x + \frac{3}{2})\) is:
\[
4x^2 + \frac{13}{2}x + \frac{3}{4}
\]

Would you like any more details or explanations?

Here are some related questions:

1. How do you multiply binomials using the FOIL method?
2. How can you simplify algebraic expressions involving fractions?
3. What are the different methods to expand binomials?
4. How do you add and subtract algebraic fractions?
5. What are some common mistakes to avoid when expanding binomials?

**Tip:** When multiplying binomials, always combine like terms to simplify the final expression.

Explore how to multiply and expand the binomials (4x + 1/2)(x + 3/2) using the distributive property, with a detailed solution suitable for students in Grades 8-10. Learn about key algebraic concepts and methods, including the distributive property and combining like terms.

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