O3/02/25 Grade 10 Factorise \( \begin{array}{ll}\text { a } x^{2}-7 x+12 & \text { () } 40 x^{2}+6 x y-18 y^{2} \\ \text { (b) } 2 x^{2}+12 x+18 & \text { (c) } 24 x^{2}-2 x y-y^{2} \\ \text { (c) } x^{2}-9 x+20 & \text { (j) } 3 x^{2}+19 x+20 \\ \text { d) } 3 x^{2}-9 x+24 & \\ \text { (e) } 2 x^{2}+5 x+3 & \\ \text { (t) } 3 x^{2}+4 x-4 & \\ \text { (g) } 5 x^{3} y+25 x^{2} y-30 x y\end{array} \)

Sure, let's factorize each of the given expressions step by step. --- ### **a) \(x^{2} - 7x + 12\)** **Factorization:** \[ x^{2} - 7x + 12 = (x - 3)(x - 4) \] **Explanation:** We look for two numbers that multiply to **12** and add up to **-7**. These numbers are **-3** and **-4**. --- ### **b) \(40x^{2} + 6xy - 18y^{2}\)** **Factorization:** \[ 40x^{2} + 6xy - 18y^{2} = 2(5x - 3y)(4x + 3y) \] **Explanation:** 1. **Factor out the Greatest Common Factor (GCF):** \(40x^{2} + 6xy - 18y^{2} = 2(20x^{2} + 3xy - 9y^{2})\) 2. **Factor the quadratic:** \(20x^{2} + 3xy - 9y^{2} = (5x - 3y)(4x + 3y)\) --- ### **c) \(2x^{2} + 12x + 18\)** **Factorization:** \[ 2x^{2} + 12x + 18 = 2(x + 3)^{2} \] **Explanation:** 1. **Factor out the GCF:** \(2x^{2} + 12x + 18 = 2(x^{2} + 6x + 9)\) 2. **Recognize a perfect square trinomial:** \(x^{2} + 6x + 9 = (x + 3)^{2}\) --- ### **d) \(24x^{2} - 2xy - y^{2}\)** **Factorization:** \[ 24x^{2} - 2xy - y^{2} = (6x + y)(4x - y) \] **Explanation:** We look for two binomials that multiply to give the original quadratic: \[ (6x + y)(4x - y) = 24x^{2} - 2xy - y^{2} \] --- ### **e) \(x^{2} - 9x + 20\)** **Factorization:** \[ x^{2} - 9x + 20 = (x - 4)(x - 5) \] **Explanation:** We look for two numbers that multiply to **20** and add up to **-9**. These numbers are **-4** and **-5**. --- ### **f) \(3x^{2} + 19x + 20\)** **Factorization:** \[ 3x^{2} + 19x + 20 = (3x + 4)(x + 5) \] **Explanation:** We find two numbers that multiply to \(3 \times 20 = 60\) and add up to **19**. These numbers are **15** and **4**. \[ 3x^{2} + 19x + 20 = 3x^{2} + 15x + 4x + 20 = (3x + 4)(x + 5) \] --- ### **g) \(3x^{2} - 9x + 24\)** **Factorization:** \[ 3x^{2} - 9x + 24 = 3(x^{2} - 3x + 8) \] **Explanation:** 1. **Factor out the GCF:** \(3x^{2} - 9x + 24 = 3(x^{2} - 3x + 8)\) 2. **Note:** The quadratic \(x^{2} - 3x + 8\) does not factor neatly with integers, so we leave it as is. --- ### **h) \(2x^{2} + 5x + 3\)** **Factorization:** \[ 2x^{2} + 5x + 3 = (2x + 3)(x + 1) \] **Explanation:** We look for two numbers that multiply to \(2 \times 3 = 6\) and add up to **5**. These numbers are **3** and **2**. \[ 2x^{2} + 5x + 3 = (2x + 3)(x + 1) \] --- ### **i) \(3x^{2} + 4x - 4\)** **Factorization:** \[ 3x^{2} + 4x - 4 = (3x - 2)(x + 2) \] **Explanation:** We look for two numbers that multiply to \(3 \times (-4) = -12\) and add up to **4**. These numbers are **6** and **-2**. \[ 3x^{2} + 4x - 4 = 3x^{2} + 6x - 2x - 4 = (3x - 2)(x + 2) \] --- ### **j) \(5x^{3}y + 25x^{2}y - 30xy\)** **Factorization:** \[ 5x^{3}y + 25x^{2}y - 30xy = 5xy(x + 6)(x - 1) \] **Explanation:** 1. **Factor out the GCF:** \(5xy\) 2. **Factor the quadratic:** \(x^{2} + 5x - 6 = (x + 6)(x - 1)\) 3. **Combine:** \(5xy(x + 6)(x - 1)\) --- These are the fully factorized forms of the given expressions. If you have any further questions or need additional explanations, feel free to ask!