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The problem is asking to find the expression for the area of the shaded region and then factor it completely.

### Step 1: Identify the areas involved

- The shaded region consists of the difference between the larger square and the smaller square in the center.
- The side length of the **larger square** is \( z \).
  So, its area is:
  \[
  \text{Area of larger square} = z^2
  \]
  
- The side length of the **smaller square** is given as \( 9 \) (in the center).
  So, its area is:
  \[
  \text{Area of smaller square} = 9^2 = 81
  \]

### Step 2: Find the area of the shaded region

The area of the shaded region is the area of the larger square minus the area of the smaller square:
\[
\text{Shaded Area} = z^2 - 81
\]

### Step 3: Factor the expression

We can factor the expression \( z^2 - 81 \) as a difference of squares:
\[
z^2 - 81 = (z - 9)(z + 9)
\]

Thus, the expression for the area of the shaded region, fully factored, is:
\[
(z - 9)(z + 9)
\]

### Final Answer:
The area of the shaded region is \( (z - 9)(z + 9) \).

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Would you like more details or have any other questions?

### Related Questions:
1. What is the difference of squares, and how is it used in factoring?
2. How do you find the area of a square if given the side length?
3. How can the formula for the area of a square be extended to rectangles?
4. What other shapes could this approach work for, such as circles or triangles?
5. How would the expression change if the inner square had a side length different from 9?

### Tip:
Whenever you encounter a difference of squares like \( a^2 - b^2 \), remember the formula: \( a^2 - b^2 = (a - b)(a + b) \). It's a very efficient way to factor such expressions!

Find an expression for the area of the shaded region, and then factor it completely.

Learn how to find the area of the shaded region between two squares and factor the expression completely. This problem involves the concept of the difference of squares, key geometric area formulas, and algebraic factoring. Suitable for students in Grades 8-10.

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