Fully factorise each of the following expressions. \( \begin{array}{ll}\text { 1. } x^{2}+7 x+12 & \text { 2. } 2 x^{2}-14 x+24 \\ \text { 3. } x^{2}+x-12 & \text { 4. } 2 x^{2}-2 x-24 \\ \text { 5. } 2 x^{2}-4 x-30 & \text { 6. } 2 x^{2}+4 x-16\end{array} \)

Here’s the step‐by‐step fully factorised form of each expression. 1. Factorise x² + 7x + 12 Look for two numbers whose product is 12 and sum is 7. The numbers 3 and 4 work: 3 × 4 = 12 and 3 + 4 = 7. Answer: (x + 3)(x + 4) 2. Factorise 2x² − 14x + 24 Step 1: Factor out the common factor 2. 2x² − 14x + 24 = 2(x² − 7x + 12) Step 2: Factorise the quadratic inside. Look for two numbers whose product is 12 and sum is −7. The numbers −3 and −4 work: (−3)(−4) = 12 and (−3) + (−4) = −7. Answer: 2(x − 3)(x − 4) 3. Factorise x² + x − 12 Look for two numbers whose product is −12 and sum is 1. The numbers 4 and −3 work: 4 × (−3) = −12 and 4 + (−3) = 1. Answer: (x + 4)(x − 3) 4. Factorise 2x² − 2x − 24 Step 1: Factor out the common factor 2. 2x² − 2x − 24 = 2(x² − x − 12) Step 2: Factorise x² − x − 12. Look for two numbers whose product is −12 and sum is −1. The numbers −4 and 3 work: (−4) × 3 = −12 and (−4) + 3 = −1. Answer: 2(x − 4)(x + 3) 5. Factorise 2x² − 4x − 30 Step 1: Factor out the common factor 2. 2x² − 4x − 30 = 2(x² − 2x − 15) Step 2: Factorise x² − 2x − 15. Look for two numbers whose product is −15 and sum is −2. The numbers −5 and 3 work: (−5) × 3 = −15 and (−5) + 3 = −2. Answer: 2(x − 5)(x + 3) 6. Factorise 2x² + 4x − 16 Step 1: Factor out the common factor 2. 2x² + 4x − 16 = 2(x² + 2x − 8) Step 2: Factorise x² + 2x − 8. Look for two numbers whose product is −8 and sum is 2. The numbers 4 and −2 work: 4 × (−2) = −8 and 4 + (−2) = 2. Answer: 2(x + 4)(x − 2) Each expression has been completely factorised.