Simplify the following expresions: \( 2 a b^{2}(4 a-3 c+7 b c) \) \( (2 a-3 b)\left(4 a^{2}+6 a b+9 b^{2}\right) \) \( (8 x y-7 z)(9 x y+3 x-5 z) \) \( (f-15 g)(6 f-4 f g-3 g) \)

Let's simplify each of the expressions step by step. 1. **For \( 2 a b^{2}(4 a-3 c+7 b c) \)**: Distribute \( 2 a b^{2} \) into the parenthesis: \[ = 2 a b^{2} \cdot 4 a - 2 a b^{2} \cdot 3 c + 2 a b^{2} \cdot 7 b c \] \[ = 8 a^{2} b^{2} - 6 a b^{2} c + 14 a b^{3} c \] Final simplified expression: \[ 8 a^{2} b^{2} + (14 a b^{3} c - 6 a b^{2} c) \] 2. **For \( (2 a-3 b)(4 a^{2}+6 a b+9 b^{2}) \)**: Use the distributive property: \[ = 2 a (4 a^{2}+6 a b+9 b^{2}) - 3 b (4 a^{2}+6 a b+9 b^{2}) \] Calculate each term: \[ = 8 a^{3} + 12 a^{2} b + 18 a b^{2} - (12 a^{2} b + 18 a b^{2} + 27 b^{3}) \] Combining like terms: \[ = 8 a^3 + 0 + 0 - 27 b^3 \] Final simplified expression: \[ 8 a^{3} - 27 b^{3} \] 3. **For \( (8 x y-7 z)(9 x y+3 x-5 z) \)**: Again, distribute: \[ = 8 x y (9 x y) + 8 x y(3 x) + 8 x y(-5 z) - 7 z(9 x y) - 7 z(3 x) - 7 z(-5 z) \] Calculating each term: \[ = 72 x^2 y^2 + 24 x^2 y - 40 x y z - 63 x y z - 21 z x + 35 z^2 \] Combine like terms: \[ = 72 x^2 y^2 + 24 x^2 y - 103 x y z + 35 z^2 \] Final simplified expression: \[ 72 x^2 y^2 + 24 x^2 y - 103 x y z + 35 z^2 \] 4. **For \( (f-15 g)(6 f-4 f g-3 g) \)**: Distribute using the distributive property: \[ = f(6 f - 4 f g - 3 g) - 15 g(6 f - 4 f g - 3 g) \] Calculating gives: \[ = 6 f^2 - 4 f^2 g - 3 f g + (-90 fg + 60 f g^2 + 45 g^2) \] Combine like terms: \[ = 6 f^2 - (4 f^2 g + 90 fg) + (60 f g^2 + 45 g^2) - 3 fg \] Final simplified expression: \[ 6 f^2 - 94 fg + 60 f g^2 + 45 g^2 \] And there you have the simplified forms for each expression!