1. **Factorise** $2c^2 - 8cm - 3cm + 12m^2$. 2. **Find the coefficient of** $n$ in $(4n + 3)(n - 5) - (2n - 3)^2$. 3. **Factorise** $12x^2 - 4xy - 5y^2$. 4. **Find the HCF of** $2x^3y^2(x + 4)$, $6x^2y(x + 4)^2$, and $8xy^3(x + 4)$. 5. **Find the HCF of** $15x^3y^2$, $25x^2y^3$, and $35x^4y$. --- ### Step 1: Factorise $2c^2 - 8cm - 3cm + 12m^2$ 1. Group terms: $$(2c^2 - 8cm) + (-3cm + 12m^2)$$ 2. Factor each group: $$2c(c - 4m) - 3m(c - 4m)$$ 3. Factor out common binomial: $$(2c - 3m)(c - 4m)$$ ### Step 2: Find coefficient of $n$ in $(4n + 3)(n - 5) - (2n - 3)^2$ 1. Expand first product: $$4n \cdot n + 4n \cdot (-5) + 3 \cdot n + 3 \cdot (-5) = 4n^2 - 20n + 3n - 15 = 4n^2 - 17n - 15$$ 2. Expand second square: $$(2n - 3)^2 = 4n^2 - 12n + 9$$ 3. Subtract: $$[4n^2 - 17n - 15] - [4n^2 - 12n + 9] = 4n^2 - 17n - 15 - 4n^2 + 12n - 9 = (-17n + 12n) + (-15 - 9) = -5n - 24$$ 4. Coefficient of $n$ is $-5$. ### Step 3: Factorise $12x^2 - 4xy - 5y^2$ 1. Multiply $12 \times (-5) = -60$ 2. Find two numbers that multiply to $-60$ and add to $-4$: $-10$ and $6$ 3. Rewrite middle term: $$12x^2 - 10xy + 6xy - 5y^2$$ 4. Group terms: $$(12x^2 - 10xy) + (6xy - 5y^2)$$ 5. Factor each group: $$2x(6x - 5y) + y(6x - 5y)$$ 6. Factor out common binomial: $$(2x + y)(6x - 5y)$$ ### Step 4: Find HCF of $2x^3y^2(x + 4)$, $6x^2y(x + 4)^2$, and $8xy^3(x + 4)$ 1. Factor each term: - $2x^3y^2(x + 4)$ - $6x^2y(x + 4)^2$ - $8xy^3(x + 4)$ 2. Find HCF of coefficients: $ ext{HCF}(2, 6, 8) = 2$ 3. Find HCF of $x$ powers: $ ext{min}(3, 2, 1) = 1$ 4. Find HCF of $y$ powers: $ ext{min}(2, 1, 3) = 1$ 5. Find HCF of $(x + 4)$ powers: $ ext{min}(1, 2, 1) = 1$ 6. Combine: $$2xy(x + 4)$$ ### Step 5: Find HCF of $15x^3y^2$, $25x^2y^3$, and $35x^4y$ 1. Find HCF of coefficients: $ ext{HCF}(15, 25, 35) = 5$ 2. Find HCF of $x$ powers: $ ext{min}(3, 2, 4) = 2$ 3. Find HCF of $y$ powers: $ ext{min}(2, 3, 1) = 1$ 4. Combine: $$5x^2y$$