Question

which model represents the factors of x² + 9x + 8?

Explanation:

Step1: Factor the quadratic expression

We need to factor \(x^{2}+9x + 8\). We look for two numbers that multiply to \(8\) (the constant term) and add up to \(9\) (the coefficient of \(x\)). The numbers are \(8\) and \(1\) since \(8\times1=8\) and \(8 + 1=9\). So \(x^{2}+9x + 8=(x + 8)(x+1)\).

Step2: Analyze the area - model representation

In an area - model for factoring a quadratic \(ax^{2}+bx + c\), the quadratic \(x^{2}+9x + 8\) can be represented as a rectangle with area \(x^{2}+9x + 8\). The rectangle can be decomposed into smaller rectangles. The \(x^{2}\) term is a square with side - length \(x\), the \(9x\) term is composed of \(9\) rectangles with dimensions \(x\times1\), and the constant \(8\) is composed of \(8\) unit squares. The factored form \((x + 8)(x + 1)\) means that the rectangle can be arranged in a way that one side has length \(x+8\) and the other has length \(x + 1\).

Answer:

The model that has a large square representing \(x^{2}\), \(9\) rectangles representing \(9x\) (with dimensions \(x\times1\)) and \(8\) unit squares, and can be arranged to show side - lengths of \(x + 8\) and \(x+1\) is the correct one. Without seeing the full options clearly, but conceptually, the model should have all positive tiles (no negative tiles) since the original quadratic \(x^{2}+9x + 8\) has all positive coefficients.