3) \( 15 m^{2}+16 m-15 \) 4) \( 15 a^{2}-8 a-12 \) 5) \( 18 a^{2}-13 a-5 \)

To factor the given quadratic expressions, we will use the method of factoring by grouping or the quadratic formula when necessary. Let's go through each expression one by one. ### 3) \( 15 m^{2}+16 m-15 \) We need to factor the quadratic expression \( 15 m^{2}+16 m-15 \). 1. Multiply the coefficient of \( m^2 \) (which is 15) by the constant term (which is -15): \[ 15 \times -15 = -225 \] 2. We need to find two numbers that multiply to -225 and add to 16. The numbers are 25 and -9. 3. Rewrite the middle term using these numbers: \[ 15 m^{2} + 25 m - 9 m - 15 \] 4. Group the terms: \[ (15 m^{2} + 25 m) + (-9 m - 15) \] 5. Factor out the common factors in each group: \[ 5m(3m + 5) - 3(3m + 5) \] 6. Now factor out the common binomial: \[ (3m + 5)(5m - 3) \] So, the factored form of \( 15 m^{2}+16 m-15 \) is: \[ (3m + 5)(5m - 3) \] ### 4) \( 15 a^{2}-8 a-12 \) Now, let's factor \( 15 a^{2}-8 a-12 \). 1. Multiply the coefficient of \( a^2 \) (which is 15) by the constant term (which is -12): \[ 15 \times -12 = -180 \] 2. We need to find two numbers that multiply to -180 and add to -8. The numbers are 12 and -15. 3. Rewrite the middle term using these numbers: \[ 15 a^{2} + 12 a - 15 a - 12 \] 4. Group the terms: \[ (15 a^{2} + 12 a) + (-15 a - 12) \] 5. Factor out the common factors in each group: \[ 3a(5a + 4) - 3(5a + 4) \] 6. Now factor out the common binomial: \[ (5a + 4)(3a - 3) \] 7. Simplifying \( 3a - 3 \): \[ 3(a - 1) \] So, the factored form of \( 15 a^{2}-8 a-12 \) is: \[ (5a + 4)(3(a - 1)) = (5a + 4)(3a - 3) \] ### 5) \( 18 a^{2}-13 a-5 \) Finally, let's factor \( 18 a^{2}-13 a-5 \). 1. Multiply the coefficient of \( a^2 \) (which is 18) by the constant term (which is -5): \[ 18 \times -5 = -90 \] 2. We need to find two numbers that multiply to -90 and add to -13. The numbers are -18 and 5. 3. Rewrite the middle term using these numbers: \[ 18 a^{2} - 18 a + 5 a - 5 \] 4. Group the terms: \[ (18 a^{2} - 18 a) + (5 a - 5) \] 5. Factor out the common factors in each group: \[ 18a(a - 1) + 5(a - 1) \] 6. Now factor out the common binomial: \[ (a - 1)(18a + 5) \] So, the factored form of \( 18 a^{2}-13 a-5 \) is: \[ (18a + 5)(a - 1) \] ### Summary of Factored Forms: 1. \( 15 m^{2}+16 m-15 = (3m + 5)(5m - 3) \) 2. \( 15 a^{2}-8 a-12 = (5a + 4)(3a - 3) \) 3. \( 18 a^{2}-13 a-5 = (18a + 5)(a