https://file.aiquickdraw.com/mathbot/file/1726139159912issde3na.jpg

To solve the problem, let's break down each part of the image:

- The larger rectangle has dimensions \( (x+3) \) and \( (7x+5) \).
- The smaller rectangle inside has dimensions \( x \) and \( (x-7) \).

Now, we'll address each question one by one:

### a. Find the area of the larger region (outer rectangle):
The area of a rectangle is given by the formula:
\[
\text{Area} = \text{length} \times \text{width}
\]
So, for the larger rectangle:
\[
\text{Area of larger region} = (x + 3)(7x + 5)
\]
Now, expand the expression:
\[
(x + 3)(7x + 5) = x(7x + 5) + 3(7x + 5)
= 7x^2 + 5x + 21x + 15
= 7x^2 + 26x + 15
\]

### b. Find the area of the smaller region (inner rectangle):
For the smaller rectangle, the area is:
\[
\text{Area of smaller region} = x(x - 7)
= x^2 - 7x
\]

### c. Find the area of the shaded region:
The shaded region is the difference between the area of the larger region and the area of the smaller region:
\[
\text{Area of shaded region} = \text{Area of larger region} - \text{Area of smaller region}
= (7x^2 + 26x + 15) - (x^2 - 7x)
\]
Now, simplify the expression:
\[
= 7x^2 + 26x + 15 - x^2 + 7x
= (7x^2 - x^2) + (26x + 7x) + 15
= 6x^2 + 33x + 15
\]

### Final Answers:
- **a. Area of the larger region**: \( 7x^2 + 26x + 15 \)
- **b. Area of the smaller region**: \( x^2 - 7x \)
- **c. Area of the shaded region**: \( 6x^2 + 33x + 15 \)

Do you need more details or have any questions about the process?

Here are 5 related questions to further explore:
1. What is the perimeter of the larger rectangle?
2. What is the perimeter of the smaller rectangle?
3. How would the areas change if the dimensions of the rectangles were doubled?
4. Can you find the value of \( x \) if the shaded area equals 100 square units?
5. How would you express the area of the shaded region as a function of \( x \)?

**Tip**: When simplifying polynomial expressions, combine like terms carefully to avoid errors.

Find the area of the larger region, the smaller region, and the shaded region based on the dimensions provided in the image.

Learn how to find the areas of a larger region, smaller region, and shaded region given a rectangular shape with algebraic expressions for the dimensions. Suitable for students in Grades 8-10.

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