Question

use a geometric model to factor ( x^2 + 3x + 2 ) by following these steps: step 1: model the trinomial by placing tiles in the product section to form a rectangle at the top left. the ( x^2 ) tile is already in the correct place.

Explanation:

Step1: Recall the trinomial form

We have the trinomial \(x^{2}+3x + 2\). For factoring a quadratic trinomial \(ax^{2}+bx + c\) (here \(a = 1\), \(b=3\), \(c = 2\)), we need to find two numbers that multiply to \(ac=1\times2 = 2\) and add up to \(b = 3\). The numbers are \(1\) and \(2\) since \(1\times2=2\) and \(1 + 2=3\).

Step2: Rewrite the middle term

We can rewrite the trinomial as \(x^{2}+x+2x + 2\).

Step3: Group and factor

Group the first two terms and the last two terms: \((x^{2}+x)+(2x + 2)\). Factor out the greatest common factor from each group: \(x(x + 1)+2(x + 1)\). Now, factor out the common binomial factor \((x + 1)\): \((x + 1)(x+2)\).

Using the geometric model (area of a rectangle), the area of the rectangle is \(x^{2}+3x + 2\). The length and width of the rectangle will be the factors. The \(x^{2}\) tile is at the top - left. We need to place \(x\) tiles and unit tiles such that the area is represented. If we consider the length as \((x + 2)\) and the width as \((x + 1)\), the area of the rectangle with length \(x + 2\) and width \(x+1\) is \((x + 1)(x + 2)=x^{2}+2x+x + 2=x^{2}+3x + 2\), which matches the given trinomial.

Answer:

The factored form of \(x^{2}+3x + 2\) is \((x + 1)(x + 2)\)