Multiply :
`2x + (1)/(2)y` and `2x - (1)/(2)y`

Answer

To multiply the expressions \(2x + \frac{1}{2}y\) and \(2x - \frac{1}{2}y\), we can use the distributive property (also known as the FOIL method for binomials). Here’s a step-by-step solution:

### Step 1: Write down the expressions
We start with the two expressions:
\[ (2x + \frac{1}{2}y)(2x - \frac{1}{2}y) \]

### Step 2: Apply the distributive property
We will multiply each term in the first expression by each term in the second expression.

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

2. **Multiply the outer terms**:
\[ 2x \cdot -\frac{1}{2}y = -x y \]

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

4. **Multiply the last terms**:
\[ \frac{1}{2}y \cdot -\frac{1}{2}y = -\frac{1}{4}y^2 \]

### Step 3: Combine all the products
Now we combine all the results from the multiplications:
\[ 4x^2 - xy + xy - \frac{1}{4}y^2 \]

### Step 4: Simplify the expression
Notice that \(-xy\) and \(xy\) cancel each other out:
\[ 4x^2 - \frac{1}{4}y^2 \]

### Final Answer
Thus, the final result of multiplying the two expressions is:
\[ 4x^2 - \frac{1}{4}y^2 \]

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