To solve this problem, we first need to understand the dimensions of the rectangle based on the given information:

1. Length of the rectangle is given as $3x$.

2. Width of the rectangle is said to be $4y$ times more than the length.

   So, the width can be expressed as: $\text{Width} = \text{Length} + 4y \times \text{Length}$

   Substituting the length $3x$ into the equation: $\text{Width} = 3x + 4y \cdot 3x$ $\text{Width} = 3x + 12xy$

Now, let's proceed to find the perimeter and area of the rectangle.

1. Perimeter of the Rectangle

The formula for the perimeter $P$ of a rectangle is: $P = 2 \times (\text{Length} + \text{Width})$

Substituting the values we have: $P = 2 \times (3x + (3x + 12xy))$ $P = 2 \times (3x + 3x + 12xy)$ $P = 2 \times (6x + 12xy)$ $P = 12x + 24xy$

2. Area of the Rectangle

The formula for the area $A$ of a rectangle is: $A = \text{Length} \times \text{Width}$

Substituting the values we have: $A = 3x \times (3x + 12xy)$ $A = 9x^2 + 36x^2y$

Therefore, the perimeter of the rectangle is $12x + 24xy$, and the area of the rectangle is $9x^2 + 36x^2y$.

Would you like more details or have any questions about this solution?

Here are some related questions to explore further:

1. What happens to the area and perimeter if $x$ and $y$ are doubled?
2. If the width is reduced by half, what would be the new perimeter and area?
3. How does the area change if $y$ is set to zero?
4. Can we express the width in terms of only $x$ if $y = 1$?
5. What are the dimensions of the rectangle if $x = 2$ and $y = 3$?

Tip: When dealing with geometry problems, it is often helpful to draw a diagram and label the dimensions to visualize the relationships more clearly.