Factorise fully 1) \( 2 x^{2}+x-3 \) 2) \( 2 x^{2}+5 x-3 \) 3) \( 2 x^{2}-22 x+20 \) 4) \( 4 x^{2}+10 x-6 \)

Sure, let's factor each of the given quadratic expressions step by step. --- ### 1) \( 2x^{2} + x - 3 \) **Step 1:** Identify coefficients: - \( a = 2 \) - \( b = 1 \) - \( c = -3 \) **Step 2:** Find two numbers that multiply to \( a \times c = 2 \times (-3) = -6 \) and add up to \( b = 1 \). The numbers are **3** and **-2** because: - \( 3 \times (-2) = -6 \) - \( 3 + (-2) = 1 \) **Step 3:** Rewrite the middle term using these numbers: \[ 2x^{2} + 3x - 2x - 3 \] **Step 4:** Factor by grouping: \[ (2x^{2} + 3x) + (-2x - 3) = x(2x + 3) -1(2x + 3) \] **Step 5:** Factor out the common binomial: \[ (x - 1)(2x + 3) \] **Final Factored Form:** \[ 2x^{2} + x - 3 = (x - 1)(2x + 3) \] --- ### 2) \( 2x^{2} + 5x - 3 \) **Step 1:** Identify coefficients: - \( a = 2 \) - \( b = 5 \) - \( c = -3 \) **Step 2:** Find two numbers that multiply to \( a \times c = 2 \times (-3) = -6 \) and add up to \( b = 5 \). The numbers are **6** and **-1** because: - \( 6 \times (-1) = -6 \) - \( 6 + (-1) = 5 \) **Step 3:** Rewrite the middle term using these numbers: \[ 2x^{2} + 6x - x - 3 \] **Step 4:** Factor by grouping: \[ (2x^{2} + 6x) + (-x - 3) = 2x(x + 3) -1(x + 3) \] **Step 5:** Factor out the common binomial: \[ (2x - 1)(x + 3) \] **Final Factored Form:** \[ 2x^{2} + 5x - 3 = (2x - 1)(x + 3) \] --- ### 3) \( 2x^{2} - 22x + 20 \) **Step 1:** Factor out the Greatest Common Factor (GCF) first. All terms are divisible by **2**: \[ 2(x^{2} - 11x + 10) \] **Step 2:** Now, factor \( x^{2} - 11x + 10 \). Find two numbers that multiply to \( 1 \times 10 = 10 \) and add up to \( -11 \). The numbers are **-10** and **-1** because: - \( -10 \times (-1) = 10 \) - \( -10 + (-1) = -11 \) **Step 3:** Rewrite and factor by grouping: \[ x^{2} - 10x - x + 10 = x(x - 10) -1(x - 10) \] **Step 4:** Factor out the common binomial: \[ (x - 1)(x - 10) \] **Final Factored Form:** \[ 2x^{2} - 22x + 20 = 2(x - 1)(x - 10) \] --- ### 4) \( 4x^{2} + 10x - 6 \) **Step 1:** Factor out the GCF first. All coefficients are divisible by **2**: \[ 2(2x^{2} + 5x - 3) \] **Step 2:** Now, factor \( 2x^{2} + 5x - 3 \). Find two numbers that multiply to \( 2 \times (-3) = -6 \) and add up to \( 5 \). The numbers are **6** and **-1** because: - \( 6 \times (-1) = -6 \) - \( 6 + (-1) = 5 \) **Step 3:** Rewrite the middle term using these numbers: \[ 2x^{2} + 6x - x - 3 \] **Step 4:** Factor by grouping: \[ (2x^{2} + 6x) + (-x - 3) = 2x(x + 3) -1(x + 3) \] **Step 5:** Factor out the common binomial: \[ (2x - 1)(x + 3) \] **Final Factored Form:** \[ 4x^{2} + 10x - 6 = 2(2x - 1)(x + 3) \] --- **Summary of All Factored Forms:** 1. \( 2x^{2} + x - 3 = (x - 1)(2x + 3) \) 2. \( 2x^{2} + 5x - 3 = (2x - 1)(x + 3) \) 3. \( 2x^{2} - 22x + 20 = 2(x - 1)(x - 10) \) 4. \( 4x^{2} + 10x - 6 = 2(2x - 1)(x + 3) \)