Katie's old bedroom was shaped like a rectangle. It had a length that was 3 times its width. When Katie's family moved, her new bedroom was also shaped like a rectangle. It was 4 feet longer and 3 feet wider than her old bedroom. If w represents the width of Katie's old bedroom, which expression represents the difference between the area of her new bedroom and the area of her old bedroom?

Sure! Let's break down the problem step by step:

1. Let's start by representing the width of Katie's old bedroom as "w".
2. Since the length of her old bedroom is 3 times its width, the length can be represented as "3w".
3. The area of her old bedroom can be calculated by multiplying the length and width, so the area is "w * 3w" or "3w^2".

Now let's move on to her new bedroom:

4. The new bedroom is 4 feet longer than the old bedroom, so the length of the new bedroom is "3w + 4".
5. The new bedroom is also 3 feet wider than the old bedroom, so the width of the new bedroom is "w + 3".
6. The area of her new bedroom can be calculated by multiplying the length and width, so the area is "(3w + 4) * (w + 3)".

Finally, to find the difference between the area of her new bedroom and the area of her old bedroom, we subtract the area of the old bedroom from the area of the new bedroom:

Area of new bedroom - Area of old bedroom = (3w + 4) * (w + 3) - 3w^2

Therefore, the expression that represents the difference between the area of her new bedroom and the area of her old bedroom is "(3w + 4) * (w + 3) - 3w^2".