1.6 Area of the original rectangle \( =(x+2)(x+3) \) \( =x^{2}+5 x+6 \)\( \quad \begin{array}{l}\text { What conclusion can you make out of the Calculation in } 1.5 \text { and } 1.6\end{array} \)

To analyze the area of the original rectangle given by the expression \( (x+2)(x+3) \), we can follow these steps: 1. **Expand the expression**: The area of the rectangle is given by the product of its length and width, which are represented by \( (x+2) \) and \( (x+3) \). \[ A = (x+2)(x+3) \] Expanding this expression: \[ A = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 \] This confirms that the area of the rectangle can be expressed as \( A = x^2 + 5x + 6 \). 2. **Conclusion from the calculations**: The calculations in 1.5 and 1.6 show that the area of the rectangle can be represented in two forms: as a product of its dimensions and as a polynomial expression. This indicates that the area is a quadratic function of \( x \). - The expression \( x^2 + 5x + 6 \) is a quadratic polynomial, which means that the area will change as \( x \) changes. - The coefficients of the polynomial provide information about the shape of the rectangle as \( x \) varies. Specifically, the leading coefficient (1) indicates that the area increases as \( x \) increases, and the constant term (6) represents the area when \( x = 0 \). In summary, the calculations demonstrate the relationship between the dimensions of the rectangle and its area, highlighting the quadratic nature of the area as a function of \( x \).