1. **Problem Statement:** Factor completely the expressions: a) $2x^2 + 10x + 12$ b) $5x^2 - 20$ --- 2. **Formula and Rules:** - First, find the Greatest Common Factor (GCF) of all terms. - Factor out the GCF. - For quadratic trinomials like $ax^2 + bx + c$, use factoring techniques such as the diamond method. - An expression is completely factored when no factor can be factored further. --- 3. **Solution for a) $2x^2 + 10x + 12$:** - Step 1: Find the GCF of $2x^2$, $10x$, and $12$. $$\text{GCF} = 2$$ - Factor out the GCF: $$2x^2 + 10x + 12 = 2(x^2 + 5x + 6)$$ - Step 2: Factor the quadratic inside the parentheses using the diamond method. We look for two numbers that multiply to $6$ (constant term) and add to $5$ (coefficient of $x$). These numbers are $2$ and $3$ because: $$2 \times 3 = 6$$ $$2 + 3 = 5$$ - Step 3: Write the factored form: $$2(x + 2)(x + 3)$$ --- 4. **Solution for b) $5x^2 - 20$:** - Step 1: Find the GCF of $5x^2$ and $-20$. $$\text{GCF} = 5$$ - Factor out the GCF: $$5x^2 - 20 = 5(x^2 - 4)$$ - Step 2: Recognize $x^2 - 4$ as a difference of squares: $$x^2 - 4 = (x - 2)(x + 2)$$ - Step 3: Write the factored form: $$5(x - 2)(x + 2)$$ --- **Final answers:** a) $2(x + 2)(x + 3)$ b) $5(x - 2)(x + 2)$