To write a polynomial that represents the area of the square based on the given dimensions and the area model, let's break it down step by step.

From your description, the side lengths of the square seem to be composed of three terms: $x$, 7, and $x$. We can organize this into an area model that represents the length and width of the square:

Length of the square = $x + 7 + x = 2x + 7$

Width of the square = $x + 7 + x = 2x + 7$

Since the area of a square is calculated by multiplying the length by the width, the area will be:

$A = (2x + 7) \times (2x + 7)$

This is a binomial squared, so we can expand it:

$A = (2x + 7)^2$

Now, let's expand the expression using the distributive property (also known as FOIL):

$A = (2x + 7)(2x + 7)$

$A = 2x(2x) + 2x(7) + 7(2x) + 7(7)$

$A = 4x^2 + 14x + 14x + 49$

Simplifying:

$A = 4x^2 + 28x + 49$

So, the polynomial representing the area of the square is:

$A = 4x^2 + 28x + 49$

Would you like to see the process in more detail or have any further questions on this?

Here are five related questions to explore further:

1. What happens if we change the side length to a different binomial expression, like $x + 5$?
2. How would you expand a binomial raised to the third power?
3. How can the distributive property be applied to polynomials with more terms?
4. What is the geometric interpretation of $4x^2 + 28x + 49$ in terms of areas?
5. How do we factor the polynomial $4x^2 + 28x + 49$ back into a binomial?

Tip: Always check if you can factor a polynomial back into a binomial form for easier calculations or simplifications.