Question

1. $x^2 + 10x + 16$

x （a box divided into four small rectangles）

Explanation:

Step1: Factor the quadratic expression

We need to factor \(x^{2}+10x + 16\). We look for two numbers that multiply to \(16\) (the constant term) and add up to \(10\) (the coefficient of the \(x\) term). The numbers \(2\) and \(8\) satisfy this because \(2\times8 = 16\) and \(2 + 8=10\).

Step2: Write the factored form

Using the numbers we found, we can factor the quadratic as \((x + 2)(x+8)\). If we were to use the area model (the box method) shown, the first term \(x^{2}\) is represented by \(x\times x\), the \(10x\) term would be split into \(2x+8x\), and the constant term \(16\) is \(2\times8\). So the box would have \(x\) and \(2\) in one part, \(x\) and \(8\) in another, etc. But the factored form of \(x^{2}+10x + 16\) is \((x + 2)(x + 8)\).

Answer:

The factored form of \(x^{2}+10x + 16\) is \((x + 2)(x + 8)\)