1. **Problem:** Factor completely the expression $x^2 + x - 20$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multiply to $ac$ and add to $b$. 3. **Work:** Here, $a=1$, $b=1$, $c=-20$. We need two numbers that multiply to $-20$ and add to $1$. These are $5$ and $-4$. 4. **Factorization:** $$x^2 + x - 20 = x^2 + 5x - 4x - 20 = (x^2 + 5x) - (4x + 20) = x(x + 5) - 4(x + 5) = (x - 4)(x + 5)$$ --- 1. **Problem:** Factor completely the expression $x^2 - 12x + 36$. 2. **Formula and rules:** Recognize perfect square trinomials: $a^2 - 2ab + b^2 = (a - b)^2$. 3. **Work:** Here, $a=1$, $b=-12$, $c=36$. Note that $36 = 6^2$ and $-12 = -2 \times 1 \times 6$. 4. **Factorization:** $$x^2 - 12x + 36 = (x - 6)^2$$ --- 1. **Problem:** Factor completely the expression $\frac{1}{4}x^2 + 2x + 3$. 2. **Formula and rules:** Multiply entire expression by 4 to clear fraction, then factor. 3. **Work:** Multiply by 4: $$4 \times \left(\frac{1}{4}x^2 + 2x + 3\right) = x^2 + 8x + 12$$ Find two numbers that multiply to $12$ and add to $8$: $6$ and $2$. 4. **Factorization:** $$x^2 + 8x + 12 = (x + 6)(x + 2)$$ Divide back by 4 (factoring out $\frac{1}{4}$): $$\frac{1}{4}x^2 + 2x + 3 = \frac{1}{4}(x + 6)(x + 2)$$ --- 1. **Problem:** Factor completely the expression $2x^2 + 12x + 18$. 2. **Formula and rules:** Factor out the greatest common factor (GCF) first. 3. **Work:** GCF is 2: $$2x^2 + 12x + 18 = 2(x^2 + 6x + 9)$$ Factor inside the parentheses: Find two numbers that multiply to $9$ and add to $6$: $3$ and $3$. 4. **Factorization:** $$2(x^2 + 6x + 9) = 2(x + 3)^2$$