The problem asks to find an expression representing the number of new tiles an employee must add to expand an $n \times n$ square floor design to the next larger square. Let $n$ represent the number of tiles along each edge of the square.

Step-by-Step Explanation:

1. The current square has an area of $n^2$, since the area of a square is the side length squared.
2. The next square will have an area of $(n+1)^2$, as it will be one tile longer on each side.
3. To find the number of new tiles needed, subtract the area of the current square from the area of the new square:

$\text{New tiles} = (n+1)^2 - n^2$

Simplifying the expression:

$(n+1)^2 = n^2 + 2n + 1$ $\text{New tiles} = (n^2 + 2n + 1) - n^2 = 2n + 1$

Thus, the number of new tiles required to expand the design is $2n + 1$.

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

Here are 5 related questions to deepen understanding:

1. How does the expression $2n + 1$ change as $n$ increases?
2. What is the total area of the new square if the original design had side length $n = 5$?
3. Can you derive the same result by considering the tiles added along the perimeter of the square?
4. How does this problem relate to the concept of differences in consecutive perfect squares?
5. What is the physical interpretation of the expression $2n + 1$ in terms of adding rows and columns of tiles?

Tip: When dealing with geometric problems, always try to break down how each dimension changes step by step.