Fully simplify each of the following expressions. \( \begin{array}{ll}\text { (1) } \frac{1}{3} x(15 x-12 y)-\frac{1}{2}(6 x-4 y) & \text { (2) }-\frac{1}{5} x(25 x-10 y)+\frac{1}{3}(12 x-3 y) \\ \begin{array}{ll}\text { (3) } \frac{1}{7} p(14 q-21 y)-\frac{1}{5} g(10 p-15 y) & \text { 4. }\left(\frac{1}{2} x-3\right)\left(4 x-\frac{1}{3}\right) \\ \text { 5. } 5\left(\frac{1}{5} x-1\right)\left(5 x-\frac{1}{5}\right) & \text { 6. }(2 x-3 y)^{2}-(x-y)(3 x+2 y) \\ \text { 7. }(2 a-3)^{2}-(3 a+4)(3 a-4)+2(3 a+1)^{2} & \text { 8. } 3(2 x-3)(x+2)-2(x-3)(x+3) \\ \text { 9. }(2 a+b)^{2}-(3 a+4 b)(2 a+7 b) & \text { 10. } 3(x-2 y)^{3} \\ \text { 11. }\left(\frac{x}{3}-2\right)\left(\frac{x}{2}-3\right) & \text { 12. }\left(2 x-\frac{3}{x}\right)^{2}\end{array}\end{array} \)

Let's simplify each expression step by step. 1. \( \frac{1}{3} x(15 x-12 y)-\frac{1}{2}(6 x-4 y) \) Simplifying the first term: \[ \frac{1}{3}(15x^2 - 12xy) = 5x^2 - 4xy \] For the second term: \[ -\frac{1}{2}(6x - 4y) = -3x + 2y \] Combining these results: \[ 5x^2 - 4xy - 3x + 2y \] 2. \( -\frac{1}{5} x(25 x-10 y)+\frac{1}{3}(12 x-3 y) \) Simplifying the first term: \[ -\frac{1}{5}(-25x^2 + 10xy) = -5x^2 + 2xy \] For the second term: \[ \frac{1}{3}(12x - 3y) = 4x - y \] Combining these: \[ -5x^2 + 2xy + 4x - y \] 3. \( \frac{1}{7} p(14 q-21 y)-\frac{1}{5} g(10 p-15 y) \) Simplifying the first term: \[ \frac{1}{7}(14pq - 21py) = 2pq - 3py \] For the second term: \[ -\frac{1}{5}(10gp - 15gy) = -2gp + 3gy \] Combining these gives: \[ 2pq - 3py - 2gp + 3gy \] 4. \( \left(\frac{1}{2} x-3\right)\left(4 x-\frac{1}{3}\right) \) Applying the distributive property: \[ \frac{1}{2} x (4x) - \frac{1}{2} x (3) - 3(4x) + 3\left(\frac{1}{3}\right) = 2x^2 - \frac{3}{2}x - 12x + 1 = 2x^2 - \frac{27}{2}x + 1 \] 5. \( 5\left(\frac{1}{5} x-1\right)\left(5 x-\frac{1}{5}\right) \) Simplifying inside the parentheses: \[ (x-5)(25x - 1) = 25x^2 - x - 125 \] Then, multiplying by 5: \[ 125x^2 - 5x - 625 \] 6. \( (2 x-3 y)^{2}-(x-y)(3 x+2 y) \) Expanding the first term: \[ 4x^2 - 12xy + 9y^2 \] For the second term: \[ - (3x^2 - 5xy - 2y^2) \] Combining: \[ (4x^2 - 3x^2) + (-12xy + 5xy) + (9y^2 + 2y^2) = x^2 - 7xy + 11y^2 \] 7. \( (2 a-3)^{2}-(3 a+4)(3 a-4)+2(3 a+1)^{2} \) The first term gives: \[ 4a^2 - 12a + 9 \] The second term simplifies to: \[ - (9a^2 - 16) = -9a^2 + 16 \] And for the last term: \[ 2(9a^2 + 6a + 1) = 18a^2 + 12a + 2 \] Combining gives: \[ (4a^2 - 9a^2 + 18a^2) + (-12a + 12a) + (9 + 16 + 2) = 13a^2 + 27 \] 8