Factorise \( 18 p^{2}-24 p q+8 q^{2}-32 \)

To factorize the expression \( 18 p^{2} - 24 p q + 8 q^{2} - 32 \), follow these steps: 1. **Factor Out the Greatest Common Factor (GCF):** All coefficients in the expression are divisible by 2: \[ 18 p^{2} - 24 p q + 8 q^{2} - 32 = 2(9 p^{2} - 12 p q + 4 q^{2} - 16) \] 2. **Recognize a Perfect Square:** The quadratic part \( 9 p^{2} - 12 p q + 4 q^{2} \) is a perfect square: \[ 9 p^{2} - 12 p q + 4 q^{2} = (3p - 2q)^2 \] So, the expression becomes: \[ 2((3p - 2q)^2 - 16) \] 3. **Apply the Difference of Squares Formula:** The expression \( (3p - 2q)^2 - 16 \) is a difference of squares: \[ (3p - 2q)^2 - 4^2 = (3p - 2q - 4)(3p - 2q + 4) \] 4. **Combine the Factors:** Incorporate the factored form back with the GCF: \[ 2(3p - 2q - 4)(3p - 2q + 4) \] **Final Factorization:** \[ 2(3p - 2q - 4)(3p - 2q + 4) \] **Answer:** After factoring, the expression is twice (3p – 2q – 4) times (3p – 2q + 4). Thus, 2(3p−2q−4)(3p−2q+4)