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The problem involves calculating the area of two rectangles, and the task is to determine which of the given expressions correctly represents the area of the shaded region.

Let's break down the problem:

### Step 1: Identify the Dimensions
1. The larger rectangle has dimensions:
   - Length: \( 4x + 6 \) cm
   - Width: \( 3x + 8 \) cm

2. The smaller rectangle (unshaded) has dimensions:
   - Length: \( 2x \) cm
   - Width: \( x + 3 \) cm

### Step 2: Calculate the Area
1. **Area of the larger rectangle**:
   \[
   \text{Area}_{\text{large}} = (4x + 6)(3x + 8)
   \]

2. **Area of the smaller rectangle**:
   \[
   \text{Area}_{\text{small}} = 2x(x + 3)
   \]

3. **Area of the shaded region**: 
   The shaded area is the area of the larger rectangle minus the area of the smaller rectangle.
   \[
   \text{Area}_{\text{shaded}} = \text{Area}_{\text{large}} - \text{Area}_{\text{small}}
   \]

### Step 3: Expand the Expressions
1. **Expanding the larger rectangle's area**:
   \[
   \text{Area}_{\text{large}} = (4x + 6)(3x + 8)
   = 12x^2 + 32x + 18x + 48
   = 12x^2 + 50x + 48
   \]

2. **Expanding the smaller rectangle's area**:
   \[
   \text{Area}_{\text{small}} = 2x(x + 3)
   = 2x^2 + 6x
   \]

3. **Subtracting the smaller area from the larger area**:
   \[
   \text{Area}_{\text{shaded}} = (12x^2 + 50x + 48) - (2x^2 + 6x)
   = 12x^2 - 2x^2 + 50x - 6x + 48
   = 10x^2 + 44x + 48
   \]

### Step 4: Compare with Given Options
The options given are:
- (a) \( 12x^2 + 45x + 34 \)
- (b) \( 12x^2 + 45x + 46 \)

It appears that neither option exactly matches the calculated expression \( 10x^2 + 44x + 48 \). This could be due to a mistake in transcription or a mismatch in the problem provided.

However, based on standard problem-solving steps, this is how the area should be calculated. If the expressions given in the options were intended to match, it would be advisable to recheck the problem or the options provided.

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Do you want more details, or do you have any questions? 

Here are 5 related questions:
1. How would the problem change if the dimensions of the rectangles were altered?
2. Can you find the shaded area for specific values of \(x\)?
3. What would happen if you rearranged the smaller rectangle within the larger one?
4. How would you approach this problem if one of the expressions was a square rather than a rectangle?
5. Could there be a mistake in the given answer options, and how would you verify it?

**Tip:** When working with algebraic expressions, carefully expand and simplify step by step to avoid errors.

Learn how to calculate the area of the shaded region between two rectangles with different dimensions. This problem involves geometry and area calculation, suitable for Grades 9-12.

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