1. **Problem:** Factorize completely the expressions given. 2. **Part a:** Factorize $6y^2 - 18xy$. - Identify the greatest common factor (GCF) of the terms: $6y^2$ and $18xy$. - GCF is $6y$. - Factor out $6y$: $$6y^2 - 18xy = 6y(y - 3x)$$ 3. **Part b:** Factorize $4m^2 - 1$. - Recognize this as a difference of squares: $a^2 - b^2 = (a - b)(a + b)$. - Here, $4m^2 = (2m)^2$ and $1 = 1^2$. - So, $$4m^2 - 1 = (2m - 1)(2m + 1)$$ 4. **Part c:** Factorize $2x^2 - x - 6$. - Use the AC method: multiply $a=2$ and $c=-6$ to get $-12$. - Find two numbers that multiply to $-12$ and add to $-1$ (the coefficient of $x$): these are $-4$ and $3$. - Rewrite the middle term: $$2x^2 - 4x + 3x - 6$$ - Factor by grouping: $$2x(x - 2) + 3(x - 2)$$ - Factor out the common binomial: $$(2x + 3)(x - 2)$$ 5. **Write as a single fraction:** $\frac{5p + 2}{3} - \frac{3p - 1}{4}$. - Find common denominator: $12$. - Rewrite each fraction: $$\frac{4(5p + 2)}{12} - \frac{3(3p - 1)}{12} = \frac{20p + 8}{12} - \frac{9p - 3}{12}$$ - Combine numerators: $$\frac{20p + 8 - 9p + 3}{12} = \frac{11p + 11}{12}$$ - Factor numerator: $$\frac{11(p + 1)}{12}$$ 6. **Coordinates of triangle ABC:** - Given vertices are: $$A(2, 2), B(6, 4), C(0, 0)$$ **Final answers:** - a. $6y(y - 3x)$ - b. $(2m - 1)(2m + 1)$ - c. $(2x + 3)(x - 2)$ - d. $\frac{11(p + 1)}{12}$ - e. $A(2, 2), B(6, 4), C(0, 0)$