1. **Problem Statement:** Factor the quadratic expressions: a) $y = x^2 + 13x + 12$ b) $y = x^2 + 8x + 12$ 2. **Recall:** Factoring means rewriting a quadratic as a product of two binomials: $$y = (x + m)(x + n)$$ where $m$ and $n$ satisfy: - $m + n = b$ (the coefficient of $x$) - $m \times n = c$ (the constant term) 3. **Factoring a) $y = x^2 + 13x + 12$: ** - Find two numbers $m$ and $n$ such that: $$m + n = 13$$ $$m \times n = 12$$ - Possible factor pairs of 12 are (1,12), (2,6), (3,4), etc. - Check sums: - $1 + 12 = 13$ (matches) - So, $m = 1$, $n = 12$ - Therefore, the factored form is: $$y = (x + 1)(x + 12)$$ 4. **Factoring b) $y = x^2 + 8x + 12$: ** - Find two numbers $m$ and $n$ such that: $$m + n = 8$$ $$m \times n = 12$$ - Factor pairs of 12: (1,12), (2,6), (3,4) - Check sums: - $2 + 6 = 8$ (matches) - So, $m = 2$, $n = 6$ - Factored form: $$y = (x + 2)(x + 6)$$ 5. **Summary:** - a) $y = (x + 1)(x + 12)$ - b) $y = (x + 2)(x + 6)$ These factored forms represent the dimensions of the area model, showing how the quadratic expression can be seen as the product of two binomials.