https://file.aiquickdraw.com/mathbot/file/1726132865952gflpmnhw.jpg

Let's solve the questions one by one based on the diagram provided:

### Part (a): Find the area of the rectangle in terms of \( x \) and \( y \).

- The dimensions of the rectangle are given as:
  - Length = \( 4x \) cm
  - Width = \( (3y + 2) \) cm

- The area \( A_{\text{rect}} \) of a rectangle is given by:
  \[
  A_{\text{rect}} = \text{Length} \times \text{Width}
  \]
  Substituting the given dimensions:
  \[
  A_{\text{rect}} = 4x \times (3y + 2)
  \]
  Expanding this expression:
  \[
  A_{\text{rect}} = 12xy + 8x \, \text{cm}^2
  \]
  **Answer for (a)**: \( 12xy + 8x \, \text{cm}^2 \)

---

### Part (b): Find the area of the triangle in terms of \( x \) and \( y \).

- The dimensions of the triangle are given as:
  - Base = \( 2x \) cm
  - Height = \( (y - 1) \) cm

- The area \( A_{\text{tri}} \) of a triangle is given by:
  \[
  A_{\text{tri}} = \frac{1}{2} \times \text{Base} \times \text{Height}
  \]
  Substituting the given dimensions:
  \[
  A_{\text{tri}} = \frac{1}{2} \times 2x \times (y - 1)
  \]
  Simplifying:
  \[
  A_{\text{tri}} = x(y - 1) = xy - x \, \text{cm}^2
  \]
  **Answer for (b)**: \( xy - x \, \text{cm}^2 \)

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### Part (c): Subtract the rectangle’s area from the triangle’s area.

To subtract the area of the rectangle from the area of the triangle, we have:
\[
\text{Difference} = A_{\text{tri}} - A_{\text{rect}}
\]
Substituting the expressions from parts (a) and (b):
\[
\text{Difference} = (xy - x) - (12xy + 8x)
\]
Expanding and simplifying:
\[
\text{Difference} = xy - x - 12xy - 8x
\]
\[
\text{Difference} = -11xy - 9x \, \text{cm}^2
\]
**Answer for (c)**: \( -11xy - 9x \, \text{cm}^2 \)

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Let me know if you'd like any further details!

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Here are 5 related questions to explore further:
1. What is the perimeter of the rectangle in terms of \( x \) and \( y \)?
2. What is the perimeter of the triangle in terms of \( x \) and \( y \)?
3. If \( x = 2 \) and \( y = 3 \), what is the numerical value of the area of the rectangle?
4. What is the total area of the rectangle and the triangle combined in terms of \( x \) and \( y \)?
5. How would the expressions change if the base of the triangle was \( 3x \) instead of \( 2x \)?

**Tip**: When dealing with geometry problems, always make sure to understand the relationship between dimensions (length, width, base, height) and formulas for area and perimeter!

The dimensions of a rectangle and a triangle are given. The rectangle has a length of 4x cm and a width of (3y + 2) cm. The triangle has a base of 2x cm and a height of (y - 1) cm. You are asked to find: (a) the area of the rectangle, (b) the area of the triangle, and (c) subtract the area of the rectangle from the area of the triangle.

Solve for the area of a rectangle and triangle in terms of x and y. The problem involves algebraic expressions and geometry formulas. Appropriate for students in Grades 8-10.

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