which model represents the factors of $x^{2}+9x + 8$?

Answer:

The second model.

Explanation:

Step1: Factor the quadratic expression

For a quadratic expression (ax^{2}+bx + c) ((a = 1), (b=9), (c = 8) in (x^{2}+9x + 8)), we use the formula (x^{2}+(m + n)x+mn=(x + m)(x + n)). We need to find two numbers (m) and (n) such that (m + n=b) and (mn=c). For (x^{2}+9x + 8), we look for two numbers that add up to (9) and multiply to (8). The numbers are (m = 1) and (n=8) since (1 + 8=9) and (1\times8 = 8), and ((x + 1)(x + 8)=x^{2}+8x+x + 8=x^{2}+9x + 8).

Step2: Analyze the area - model (rectangle model for factoring)

In the area - model (rectangle model) for factoring a quadratic (x^{2}+bx + c=(x + m)(x + n)), the length and width of the rectangle represent the factors. The first model has negative tiles (the blue (-) tiles), but in (x^{2}+9x + 8=(x + 1)(x + 8)) there are no negative terms. The second model has the correct combination of (x) - tiles and (+) (unit) tiles. The dimensions of the rectangle (if we consider the side - lengths formed by the sum of (x) and unit tiles) correspond to ((x + 1)) and ((x + 8)). The area of the rectangle (sum of the areas of all sub - rectangles: (x^{2}+8x+x + 8)) is (x^{2}+9x + 8).