From the image, it looks like the task involves finding polynomial expressions to represent the length and width of a corn field, which is part of a farm that uses polynomial identities. Here's a step-by-step guide to solving the problem:

1. Identify Polynomial Expressions:

   • For the pumpkin: $x^2 + 10x$
   • For the beans: $x^2 + 5x + 6$
   • For the corn: $x^2 - 4$
   • For the cabbage: $x^2 - 2x - 8$

2. Determine Length and Width of the Corn Field:

   • The corn field is specifically represented by the expression $x^2 - 4$.
   • To find its length and width, you can factorize the expression:
     • $x^2 - 4 = (x - 2)(x + 2)$
   • Therefore, the length and width of the corn field can be represented as $(x - 2)$ and $(x + 2)$.

3. Showing Work for Factorization:

   • Start with the given polynomial $x^2 - 4$.
   • Recognize it as a difference of squares: $a^2 - b^2 = (a - b)(a + b)$.
   • Here, $a = x$ and $b = 2$, so it becomes $(x - 2)(x + 2)$.

4. Placement on the Farm Diagram:

   • Length: $x + 2$
   • Width: $x - 2$

Would you like further details or have any questions about this? Here are some related questions:

1. How do you factor other types of polynomials, such as trinomials?
2. What are polynomial identities, and how can they be used in real-life applications?
3. How does recognizing a pattern like the difference of squares help simplify expressions?
4. Can you explain how to verify that the factors of a polynomial are correct?
5. How can polynomial expressions represent other areas in a farm setup?

Tip: When factorizing polynomials, always look for common patterns such as difference of squares, perfect square trinomials, or factoring by grouping to simplify the process.