https://file.aiquickdraw.com/mathbot/file/1727350888348v62ywa3i.jpg

The problem in the image is about a rectangle with:

- **Width**: \( 4a \)
- **Area**: \( 24a^2 + 8a \)

We need to solve the following:

### Part (a) - Find the length of the rectangle.
The formula for the area of a rectangle is:
\[
\text{Area} = \text{Length} \times \text{Width}
\]
Here, the area is given as \( 24a^2 + 8a \), and the width is \( 4a \).

So, using the formula:
\[
24a^2 + 8a = \text{Length} \times 4a
\]

To find the length, divide both sides by \( 4a \):
\[
\text{Length} = \frac{24a^2 + 8a}{4a}
\]
Now, simplify:
\[
\text{Length} = \frac{24a^2}{4a} + \frac{8a}{4a} = 6a + 2
\]
So, the length of the rectangle is \( 6a + 2 \).

### Part (b) - Find the perimeter of the rectangle.
The formula for the perimeter of a rectangle is:
\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width})
\]
Substitute the values of length \( 6a + 2 \) and width \( 4a \):
\[
\text{Perimeter} = 2 \times ((6a + 2) + 4a) = 2 \times (10a + 2) = 20a + 4
\]
Thus, the perimeter of the rectangle is \( 20a + 4 \).

---

Would you like further clarification on any step or more details?

Here are five related questions to expand your understanding:
1. How would the length change if the width is \( 5a \) instead of \( 4a \)?
2. What is the relationship between the length and width in this particular rectangle?
3. If the area formula was different, such as \( 30a^2 + 10a \), how would the length and perimeter change?
4. How would you solve for the dimensions of the rectangle if only the perimeter was given?
5. What role does the variable \( a \) play in changing the size of the rectangle?

**Tip**: Always break down complex algebraic expressions step-by-step to simplify and solve equations easily.

Cho hình chữ nhật có chiều rộng là 4a và diện tích là 24a^2 + 8a. Tìm chiều dài và chu vi của hình chữ nhật.

Solve a rectangle problem where the width is 4a and the area is 24a^2 + 8a. Learn to find the length and perimeter using algebraic methods suitable for Grades 7-9.

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