https://r2.erweima.ai/mathbot/image/1726154184536sebj0djs.jpg

Let's analyze the provided image, which includes two tasks involving areas of rectangles.

### Part a) 
The problem asks for an explanation of how you can calculate the area of the yellow-shaded rectangle using the two given expressions.

- The rectangle has two parts:
  1. The first part has dimensions \(5 \times x\).
  2. The second part has dimensions \(5 \times 3\).

The total area of the yellow region can be calculated in two ways:
1. **Without brackets**: Add the two individual areas.
   \[
   5 \cdot x + 5 \cdot 3
   \]
2. **With brackets**: Factor the expression to group terms:
   \[
   5 \cdot (x + 3)
   \]
Both approaches give the same result, as the distributive property of multiplication applies here.

### Part b)
This part asks for an expression both without and with brackets to find the area of the yellow regions in two new shapes.

#### 1. Figure A:
- Dimensions: \(x \times 8\) (full height) and \(2 \times 8\) (smaller rectangle)
- Without brackets: Calculate individual areas and add:
  \[
  x \cdot 8 + 2 \cdot 8 = 8x + 16
  \]
- With brackets: Factor out 8:
  \[
  8 \cdot (x + 2)
  \]

#### 2. Figure B:
- Dimensions: \(x \times 6\) (upper rectangle) and \(7 \times 6\) (lower rectangle)
- Without brackets: Calculate individual areas and add:
  \[
  x \cdot 6 + 7 \cdot 6 = 6x + 42
  \]
- With brackets: Factor out 6:
  \[
  6 \cdot (x + 7)
  \]

This shows how both forms (without and with brackets) provide equivalent ways to calculate the total area.

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Would you like further details or clarification on any part of the solution?

### Follow-up questions:
1. What is the advantage of using factored expressions with brackets in area calculations?
2. Can you explain the distributive property using a different numerical example?
3. How would you calculate the area if the dimensions involved decimals or fractions?
4. How does factoring simplify solving equations in algebra?
5. Can you derive an expression for the area if one of the rectangles had a variable width as well?

### Tip:
When simplifying expressions, always look for common factors to help you re-write the equation in a more compact form.

Calculate the area of the yellow-shaded regions for various rectangles. Express the area using both a term without brackets and a term with brackets.

Learn how to calculate the area of rectangles and express the result in two ways: using a term with brackets and without brackets. The problem covers algebraic manipulation using the distributive property. Suitable for Grades 7-9.

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