To solve this problem, let’s break it down into two parts: creating an area model and writing an expression for the area of the rectangle.

Step 1: Setting Up the Area Model

The dimensions of the rectangle are:

• Length = $4x + 1$
• Width = $x + 2$

In an area model, we break down each term in these expressions and multiply them to find the area of each part. Let’s divide the expressions as follows:

• For $4x + 1$: we’ll separate this into $4x$ and $1$.
• For $x + 2$: we’ll separate this into $x$ and $2$.

Step 2: Drawing the Area Model

To illustrate, we’ll set up a grid to represent the area model, with each part corresponding to the multiplication of these terms. The four sections will represent:

1. $4x \times x$
2. $4x \times 2$
3. $1 \times x$
4. $1 \times 2$

The area model is structured as follows:

\hline 4x \times x & 4x \times 2 \\ \hline 1 \times x & 1 \times 2 \\ \hline \end{array}$$ ### Step 3: Calculating Each Part of the Area Now, we calculate each of the four sections: 1. $$4x \times x = 4x^2$$ 2. $$4x \times 2 = 8x$$ 3. $$1 \times x = x$$ 4. $$1 \times 2 = 2$$ ### Step 4: Writing the Expression for the Total Area To find the area of the rectangle, we add up all four parts: $$\text{Area} = 4x^2 + 8x + x + 2$$ Simplifying, we combine like terms: $$\text{Area} = 4x^2 + 9x + 2$$ ### Final Answer The expression for the area of the rectangle is: $$4x^2 + 9x + 2$$ Would you like a detailed diagram or further clarification? ### Further Questions to Consider: 1. What would the area expression look like if the dimensions of the rectangle changed? 2. How would the expression simplify if $$x$$ had a specific value, say $$x = 3$$? 3. How can this area model be used to understand polynomial multiplication better? 4. If the rectangle’s width was doubled, how would the area expression change? 5. How does this area model relate to factoring quadratic expressions? ### Tip: When using an area model, always label each part clearly. This helps keep track of each term and makes combining terms easier.