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To solve this problem, we need to calculate the area of the rectangle. The area of a rectangle is given by multiplying its width (ancho) and length (largo).

### Step 1: Expressions for Width and Length
- Width: \( \frac{2}{3}x + 3 \)
- Length: \( \frac{1}{2}x - 1 \)

### Step 2: Multiply the Width and Length

\[
\left( \frac{2}{3}x + 3 \right) \times \left( \frac{1}{2}x - 1 \right)
\]

We will use the distributive property (FOIL method) to expand this:

\[
= \left( \frac{2}{3}x \right) \times \left( \frac{1}{2}x \right) + \left( \frac{2}{3}x \right) \times (-1) + 3 \times \left( \frac{1}{2}x \right) + 3 \times (-1)
\]

### Step 3: Perform Multiplications
1. \( \left( \frac{2}{3}x \right) \times \left( \frac{1}{2}x \right) = \frac{2}{3} \times \frac{1}{2}x^2 = \frac{1}{3}x^2 \)
2. \( \left( \frac{2}{3}x \right) \times (-1) = -\frac{2}{3}x \)
3. \( 3 \times \left( \frac{1}{2}x \right) = \frac{3}{2}x \)
4. \( 3 \times (-1) = -3 \)

### Step 4: Combine Terms
Now, sum up all the terms:
\[
\frac{1}{3}x^2 + \left( -\frac{2}{3}x + \frac{3}{2}x \right) - 3
\]

### Step 5: Combine the Linear Terms
We need a common denominator to combine the linear terms:
\[
-\frac{2}{3}x + \frac{3}{2}x = \left( -\frac{4}{6}x + \frac{9}{6}x \right) = \frac{5}{6}x
\]

### Step 6: Final Expression
So, the final expression for the area is:
\[
\frac{1}{3}x^2 + \frac{5}{6}x - 3
\]

### Step 7: Match with the Choices
The correct answer is **d.** \( \frac{1}{3}x^2 + \frac{5}{6}x - 3 \).

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Would you like more details or clarifications on any of the steps?

Here are some related questions to deepen your understanding:
1. How do you factor a quadratic expression like the one found in this problem?
2. What are the properties of the distributive property used in this expansion?
3. How do you find the common denominator for adding fractions with different denominators?
4. What would the area of the rectangle be if \( x = 6 \)?
5. Can you explain why the distributive property is necessary for this calculation?

**Tip**: Always check if terms can be simplified before performing operations when dealing with algebraic expressions.

Andrea necesita calcular el área de un rectángulo. El ancho está dado por la expresión \(\frac{2}{3}x + 3\) y el largo por \(\frac{1}{2}x - 1\). ¿Cuál es la expresión para el área del rectángulo?

Learn how to calculate the area of a rectangle given its width and length as algebraic expressions. This problem involves multiplying binomials and using the distributive property to find the correct quadratic expression. Suitable for students in Grades 8-10.

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