6) \( 20 m^{2}+44 m-15 \) 7) \( 20 n^{2}-9 n-20 \) 8) \( 20 a^{2}-7 a-40 \)

To factor the given quadratic expressions, we will use the method of factoring by grouping or the quadratic formula where necessary. Let's go through each expression one by one. ### 6) \( 20 m^{2}+44 m-15 \) To factor \( 20 m^{2}+44 m-15 \), we look for two numbers that multiply to \( 20 \times -15 = -300 \) and add to \( 44 \). The numbers that work are \( 60 \) and \( -5 \). Now we can rewrite the middle term: \[ 20 m^{2} + 60 m - 5 m - 15 \] Next, we group the terms: \[ (20 m^{2} + 60 m) + (-5 m - 15) \] Factoring out the common factors in each group: \[ 20 m(m + 3) - 5(m + 3) \] Now we can factor out the common binomial factor: \[ (20 m - 5)(m + 3) \] We can simplify \( 20 m - 5 \) to \( 5(4 m - 1) \): \[ 5(4 m - 1)(m + 3) \] ### Final Factorization: \[ 5(4 m - 1)(m + 3) \] --- ### 7) \( 20 n^{2}-9 n-20 \) For \( 20 n^{2}-9 n-20 \), we look for two numbers that multiply to \( 20 \times -20 = -400 \) and add to \( -9 \). The numbers that work are \( 16 \) and \( -25 \). Now we can rewrite the middle term: \[ 20 n^{2} + 16 n - 25 n - 20 \] Next, we group the terms: \[ (20 n^{2} + 16 n) + (-25 n - 20) \] Factoring out the common factors in each group: \[ 4 n(5 n + 4) - 5(5 n + 4) \] Now we can factor out the common binomial factor: \[ (4 n - 5)(5 n + 4) \] ### Final Factorization: \[ (4 n - 5)(5 n + 4) \] --- ### 8) \( 20 a^{2}-7 a-40 \) For \( 20 a^{2}-7 a-40 \), we look for two numbers that multiply to \( 20 \times -40 = -800 \) and add to \( -7 \). The numbers that work are \( 25 \) and \( -32 \). Now we can rewrite the middle term: \[ 20 a^{2} + 25 a - 32 a - 40 \] Next, we group the terms: \[ (20 a^{2} + 25 a) + (-32 a - 40) \] Factoring out the common factors in each group: \[ 5 a(4 a + 5) - 8(4 a + 5) \] Now we can factor out the common binomial factor: \[ (5 a - 8)(4 a + 5) \] ### Final Factorization: \[ (5 a - 8)(4 a + 5) \] --- ### Summary of Factorizations: 1. \( 20 m^{2}+44 m-15 = 5(4 m - 1)(m + 3) \) 2. \( 20 n^{2}-9 n-20 = (4 n - 5)(5 n + 4) \) 3. \( 20 a^{2}-7 a-40 = (5 a - 8)(4 a + 5) \)