Use a geometric model to factor 3x² + 4x + 1 by following these steps: Model the trinomial by placing tiles to form a rectangle in the top left corner of the Product section.

Steps to Solve

1. Identify the Terms of the Polynomial

Start with the polynomial $3x^2 + 4x + 1$. Identify the coefficients:

• The coefficient of $x^2$ is 3,
• The coefficient of $x$ is 4,
• The constant term is 1.

2. Create a Rectangle Representation

Begin to model the trinomial as a rectangle. The area of the rectangle represents the polynomial. The length can be associated with the term $3x^2$ and the width with $4x + 1$.

3. Divide the Area into Sections

To factor this polynomial, divide the rectangle:

• The area represented by $3x^2$ can be expressed as $3x \cdot x$.
• The remaining area must correspond to $4x + 1$.

4. Determine Dimensions

Assume:

• Factor 1: $3x + 1$
• Factor 2: $x + 1$

This means the rectangle can be split accordingly:

• The area of Factor 1 is $3x + 1$ and Factor 2 is $x + 1$.

5. Verify the Factors

To confirm the factors, multiply: $$ (3x + 1)(x + 1) = 3x^2 + 3x + 1x + 1 = 3x^2 + 4x + 1 $$

Confirming that these factors yield the original polynomial.