Thank you for the correction. Let's recalculate the areas based on the updated dimensions.

Updated Problem:

1. The larger rectangle has dimensions:

   • Length: $4x + 5$ cm
   • Width: $3x + 8$ cm

2. The smaller rectangle (unshaded) has dimensions:

   • Length: $2x$ cm
   • Width: $x + 3$ cm

Step 1: Calculate the Areas

1. Area of the larger rectangle: $\text{Area}_{\text{large}} = (4x + 5)(3x + 8)$

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 2: Expand the Expressions

1. Expanding the larger rectangle's area: = 12x^2 + 32x + 15x + 40 = 12x^2 + 47x + 40$$
2. Expanding the smaller rectangle's area: = 2x^2 + 6x$$
3. Subtracting the smaller area from the larger area: = 12x^2 - 2x^2 + 47x - 6x + 40 = 10x^2 + 41x + 40$$

Step 3: Compare with Given Options

The options given are:

• (a) $12x^2 + 45x + 34$
• (b) $12x^2 + 45x + 46$

Since neither of these exactly matches the calculated area of $10x^2 + 41x + 40$, there may still be a mismatch between the correct expression and the options provided. However, the process above shows the correct algebraic steps.

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Do you have any questions, or would you like more details?

Here are 5 related questions:

1. What happens if you change the dimensions of the smaller rectangle?
2. How would the area change if one side of the smaller rectangle doubled?
3. Can you calculate the area for a specific value of $x$?
4. What is the impact of errors in transcription on solving such problems?
5. How would you explain the difference between the correct area and the given options?

Tip: Always verify given answer choices by plugging in a value for $x$ to check if the simplified expression holds.