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

Let's expand and simplify each of the given expressions step by step. --- ### **(1)** \( 2x(3x - 4y) - (7x^{2} - 2xy) \) **Step 1: Distribute the terms inside the parentheses.** \[ 2x(3x) + 2x(-4y) - 7x^{2} + 2xy \] **Step 2: Perform the multiplications.** \[ 6x^{2} - 8xy - 7x^{2} + 2xy \] **Step 3: Combine like terms.** \[ (6x^{2} - 7x^{2}) + (-8xy + 2xy) = -x^{2} - 6xy \] **Final Answer:** \[ -x^{2} - 6xy \] --- ### **(2)** \( (5y + 1)^{2} - (3y + 4)(2 - 3y) \) **Step 1: Expand \( (5y + 1)^{2} \).** \[ (5y)^{2} + 2 \cdot 5y \cdot 1 + 1^{2} = 25y^{2} + 10y + 1 \] **Step 2: Expand \( (3y + 4)(2 - 3y) \).** \[ 3y \cdot 2 + 3y \cdot (-3y) + 4 \cdot 2 + 4 \cdot (-3y) = 6y - 9y^{2} + 8 - 12y \] \[ = -9y^{2} - 6y + 8 \] **Step 3: Subtract the second expansion from the first.** \[ 25y^{2} + 10y + 1 - (-9y^{2} - 6y + 8) = 25y^{2} + 10y + 1 + 9y^{2} + 6y - 8 \] **Step 4: Combine like terms.** \[ (25y^{2} + 9y^{2}) + (10y + 6y) + (1 - 8) = 34y^{2} + 16y - 7 \] **Final Answer:** \[ 34y^{2} + 16y - 7 \] --- ### **(3)** \( (2x + y)^{2} - (3x - 2y)^{2} + (x - 4y)(x + 4y) \) **Step 1: Expand each squared term and the product.** \[ (2x + y)^{2} = 4x^{2} + 4xy + y^{2} \] \[ (3x - 2y)^{2} = 9x^{2} - 12xy + 4y^{2} \] \[ (x - 4y)(x + 4y) = x^{2} - 16y^{2} \quad (\text{Difference of squares}) \] **Step 2: Substitute the expansions into the original expression.** \[ (4x^{2} + 4xy + y^{2}) - (9x^{2} - 12xy + 4y^{2}) + (x^{2} - 16y^{2}) \] **Step 3: Distribute the negative sign and combine like terms.** \[ 4x^{2} + 4xy + y^{2} - 9x^{2} + 12xy - 4y^{2} + x^{2} - 16y^{2} \] \[ (4x^{2} - 9x^{2} + x^{2}) + (4xy + 12xy) + (y^{2} - 4y^{2} - 16y^{2}) = -4x^{2} + 16xy - 19y^{2} \] **Final Answer:** \[ -4x^{2} + 16xy - 19y^{2} \] --- ### **(4)** \( x^{6} + (x^{3} - 3y)(x^{3} + 3y) \) **Step 1: Recognize the product as a difference of squares.** \[ (x^{3} - 3y)(x^{3} + 3y) = (x^{3})^{2} - (3y)^{2} = x^{6} - 9y^{2} \] **Step 2: Add \( x^{6} \) to the expanded product.** \[ x^{6} + (x^{6} - 9y^{2}) = 2x^{6} - 9y^{2} \] **Final Answer:** \[ 2x^{6} - 9y^{2} \] --- **Summary of All Results:** 1. **(1)** \( -x^{2} - 6xy \) 2. **(2)** \( 34y^{2} + 16y - 7 \) 3. **(3)** \( -4x^{2} + 16xy - 19y^{2} \) 4. **(4)** \( 2x^{6} - 9y^{2} \)