Expand and simplify: \( \begin{array}{ll}\text { (1) } 2 x(3 x-4 y)-\left(7 x^{2}-2 x y\right) & \text { (2) } \quad(5 y+1)^{2}-(3 y+4)(2-3 y) \\ \text { (3) }(2 x+y)^{2}-(3 x-2 y)^{2}+(x-4 y)(x+4 y) \\ \text { (4) } x^{6}+\left(x^{3}-3 y\right)\left(x^{3}+3 y\right) & \text { (5) } \quad(3 a+b)(3 a-b)(2 a+5 b)\end{array} \)

For the first expression, \( 2x(3x - 4y) - (7x^2 - 2xy) \), let's distribute the \(2x\) first: \[ 2x(3x) - 2x(4y) - 7x^2 + 2xy = 6x^2 - 8xy - 7x^2 + 2xy \] Now, combining like terms gives: \[ (6x^2 - 7x^2) + (-8xy + 2xy) = -x^2 - 6xy \] For the second expression, \((5y + 1)^2 - (3y + 4)(2 - 3y)\): First, expand both parts: \[ (5y + 1)^2 = 25y^2 + 10y + 1 \] and \[ (3y + 4)(2 - 3y) = 6y - 9y^2 + 8 - 12y = -9y^2 - 6y + 8 \] Now, putting everything together: \[ 25y^2 + 10y + 1 - (-9y^2 - 6y + 8) = 25y^2 + 10y + 1 + 9y^2 + 6y - 8 \] Combining like terms yields: \[ (25y^2 + 9y^2) + (10y + 6y) + (1 - 8) = 34y^2 + 16y - 7 \] For expression (3), \((2x + y)^2 - (3x - 2y)^2 + (x - 4y)(x + 4y)\): First, expanding the squares gives: \[ (2x + y)^2 = 4x^2 + 4xy + y^2 \] and \[ (3x - 2y)^2 = 9x^2 - 12xy + 4y^2 \] The difference of these results in: \[ 4x^2 + 4xy + y^2 - (9x^2 - 12xy + 4y^2) = 4x^2 + 4xy + y^2 - 9x^2 + 12xy - 4y^2 \] which simplifies to: \[ (-5x^2 + 16xy - 3y^2) \] Next, expand \((x - 4y)(x + 4y)\): \[ x^2 - 16y^2 \] Adding this to the previous result, we have: \[ -5x^2 + 16xy - 3y^2 + x^2 - 16y^2 = -4x^2 + 16xy - 19y^2 \] Finally, for expression (4), \(x^6 + (x^3 - 3y)(x^3 + 3y)\): Using the difference of squares: \[ = x^6 + (x^6 - 9y^2) = 2x^6 - 9y^2 \] And lastly, for expression (5): \[ (3a + b)(3a - b)(2a + 5b) = (9a^2 - b^2)(2a + 5b) \] Distributing gives: \[ 18a^3 + 45a^2b - 2ab^2 - 5b^3 \] So the simplified expressions are: 1. \( -x^2 - 6xy \) 2. \( 34y^2 + 16y - 7 \) 3. \( -4x^2 + 16xy - 19y^2 \) 4. \( 2x^6 - 9y^2 \) 5. \( 18a^3 + 45a^2b - 2ab^2 - 5b^3 \)