Let's analyze the problem step by step:

Given:

1. Original dimensions:

   • Length: $6x$ meters
   • Width: $6x$ meters

2. Changes to dimensions:

   • Length is increased by 15 meters: $6x + 15$
   • Width is decreased by 15 meters: $6x - 15$

3. New area expression: The new area of the building will be the product of the new length and width: $A = \text{New Length} \times \text{New Width}$ Substituting the expressions: $A = (6x + 15)(6x - 15)$

4. Simplify using the difference of squares: $A = (6x)^2 - (15)^2$ $A = 36x^2 - 225$

Final Answer:

The expression that represents the area for the new design is: $A = 36x^2 - 225$

If you'd like further explanation or have any questions, let me know!

Here are 5 related questions to practice with:

1. What happens to the new area if $x = 5$?
2. How would the formula change if the length were increased by 10 meters instead of 15?
3. What is the perimeter of the new design in terms of $x$?
4. If the width decrease was only 10 meters, how would the expression for the new area look?
5. How can you determine the original area of the building?

Tip:

Always check for opportunities to simplify expressions, such as using the difference of squares, when dealing with products like $(a+b)(a-b)$.