Multiply. \[ \left(8 a^{2} b-9 a b+3 b^{2}\right)(a b-9 b+a) \] \( \left(8 a^{2} b-9 a b+3 b^{2}\right)(a b-9 b+a)=\square \) (Simplify your answer.)

To multiply the given expressions, we will use the distributive property (also known as the FOIL method for binomials). We start with: \[ \left(8 a^{2} b - 9 a b + 3 b^{2}\right)(a b - 9 b + a) \] Distributing each term from the first polynomial to each term in the second: 1. \( 8 a^{2} b \cdot a b = 8 a^{3} b^{2} \) 2. \( 8 a^{2} b \cdot (-9 b) = -72 a^{2} b^{2} \) 3. \( 8 a^{2} b \cdot a = 8 a^{3} b \) 4. \( -9 a b \cdot a b = -9 a^{2} b^{2} \) 5. \( -9 a b \cdot (-9 b) = 81 a b^{2} \) 6. \( -9 a b \cdot a = -9 a^{2} b \) 7. \( 3 b^{2} \cdot a b = 3 a b^{3} \) 8. \( 3 b^{2} \cdot (-9 b) = -27 b^{3} \) 9. \( 3 b^{2} \cdot a = 3 a b^{2} \) Now we combine all these results together: \[ 8 a^{3} b^{2} - 72 a^{2} b^{2} + 8 a^{3} b - 9 a^{2} b^{2} + 81 a b^{2} - 9 a^{2} b + 3 a b^{3} - 27 b^{3} + 3 a b^{2} \] Next, we combine like terms: 1. \( a^3 b^2: 8 a^3 b^2 \) 2. \( a^2 b^2: -72 a^2 b^2 - 9 a^2 b^2 = -81 a^2 b^2 \) 3. \( a^3 b: 8 a^3 b \) 4. \( a^2 b: -9 a^2 b \) 5. \( ab^2: 81 ab^2 + 3 ab^2 = 84 ab^2 \) 6. \( ab^3: 3 ab^3 \) 7. \( b^3: -27 b^3 \) The final combined expression is: \[ 8 a^{3} b + (8 a^{3} b^2) + 84 a b^{2} - 27 b^{3} - 81 a^{2} b^{2} + 3 a b^{3} \] This simplifies down to: \[ 8 a^{3} b + (-81 a^{2} b^{2} + 84 a b^{2} - 27 b^{3}) + 3 ab^{3} \] So putting everything together gives us the final answer: \[ \boxed{8 a^{3} b + (-81 a^{2} b^{2} + 84 a b^{2} + 3 a b^{3} - 27 b^{3})} \]