factorise fully a) \( x^{2}+7 x+12 \) \( = \) b) \( x^{2}-4 x-12 \) \( = \) c) \( x^{2}+4 x-12 \) \( = \) d) \( a^{2}-9 a+20 \)

Sure, let's factorize each of the given quadratic expressions step by step. --- ### a) \( x^{2} + 7x + 12 \) **Step 1:** Look for two numbers that multiply to **12** (the constant term) and add up to **7** (the coefficient of \( x \)). - Numbers: **3** and **4** \( 3 \times 4 = 12 \) \( 3 + 4 = 7 \) **Factorization:** \[ x^{2} + 7x + 12 = (x + 3)(x + 4) \] --- ### b) \( x^{2} - 4x - 12 \) **Step 1:** Find two numbers that multiply to **-12** and add up to **-4**. - Numbers: **-6** and **2** \( -6 \times 2 = -12 \) \( -6 + 2 = -4 \) **Factorization:** \[ x^{2} - 4x - 12 = (x - 6)(x + 2) \] --- ### c) \( x^{2} + 4x - 12 \) **Step 1:** Identify two numbers that multiply to **-12** and add up to **4**. - Numbers: **6** and **-2** \( 6 \times (-2) = -12 \) \( 6 + (-2) = 4 \) **Factorization:** \[ x^{2} + 4x - 12 = (x + 6)(x - 2) \] --- ### d) \( a^{2} - 9a + 20 \) **Step 1:** Find two numbers that multiply to **20** and add up to **-9**. - Numbers: **-4** and **-5** \( -4 \times (-5) = 20 \) \( -4 + (-5) = -9 \) **Factorization:** \[ a^{2} - 9a + 20 = (a - 4)(a - 5) \] --- **Summary of Factorizations:** a) \( x^{2} + 7x + 12 = (x + 3)(x + 4) \) b) \( x^{2} - 4x - 12 = (x - 6)(x + 2) \) c) \( x^{2} + 4x - 12 = (x + 6)(x - 2) \) d) \( a^{2} - 9a + 20 = (a - 4)(a - 5) \)