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.