1. \( 25 a^{2}-16 \) 3. \( 2 a^{2}+2 b^{2} \) 5. \( x^{2} y^{2}-1 \) 7. \( 28 x^{3}-63 x b^{2} \) 9. \( (2 x+3 y)^{2}-9 \) 11. \( 4 x^{2}-(3 x-2 y)^{2} \) 13. \( 25 a^{2}(a-3 b)+9(3 b-a) \)

Sure, let's factor each of the given expressions step by step. --- ### **1. \( 25 a^{2} - 16 \)** This is a **difference of squares**, which can be factored using the formula: \[ A^2 - B^2 = (A - B)(A + B) \] Here, \( A = 5a \) and \( B = 4 \). **Factored Form:** \[ 25 a^{2} - 16 = (5a - 4)(5a + 4) \] --- ### **3. \( 2 a^{2} + 2 b^{2} \)** First, factor out the common factor of **2**: \[ 2 a^{2} + 2 b^{2} = 2(a^{2} + b^{2}) \] The expression \( a^{2} + b^{2} \) does not factor further over the real numbers. **Factored Form:** \[ 2 a^{2} + 2 b^{2} = 2(a^{2} + b^{2}) \] --- ### **5. \( x^{2} y^{2} - 1 \)** This is also a **difference of squares**. Let \( A = xy \) and \( B = 1 \). **Factored Form:** \[ x^{2} y^{2} - 1 = (xy - 1)(xy + 1) \] --- ### **7. \( 28 x^{3} - 63 x b^{2} \)** First, factor out the **greatest common factor (GCF)**, which is \( 7x \): \[ 28 x^{3} - 63 x b^{2} = 7x(4x^{2} - 9b^{2}) \] Notice that \( 4x^{2} - 9b^{2} \) is a **difference of squares** where \( A = 2x \) and \( B = 3b \). **Fully Factored Form:** \[ 28 x^{3} - 63 x b^{2} = 7x(2x - 3b)(2x + 3b) \] --- ### **9. \( (2x + 3y)^{2} - 9 \)** Recognize this as a **difference of squares**, with \( A = (2x + 3y) \) and \( B = 3 \). **Factored Form:** \[ (2x + 3y)^{2} - 9 = (2x + 3y - 3)(2x + 3y + 3) \] --- ### **11. \( 4x^{2} - (3x - 2y)^{2} \)** This is a **difference of squares**, where \( A = 2x \) and \( B = (3x - 2y) \). **Factored Form:** \[ 4x^{2} - (3x - 2y)^{2} = (2x - (3x - 2y))(2x + (3x - 2y)) \] Simplify the factors: \[ = (-x + 2y)(5x - 2y) = (2y - x)(5x - 2y) \] --- ### **13. \( 25 a^{2}(a - 3 b) + 9(3 b - a) \)** First, rewrite the expression to identify a common factor: \[ 25 a^{2}(a - 3b) + 9(3b - a) = 25a^{2}(a - 3b) - 9(a - 3b) \] Factor out the common term \( (a - 3b) \): \[ = (a - 3b)(25a^{2} - 9) \] Notice that \( 25a^{2} - 9 \) is a **difference of squares**, where \( A = 5a \) and \( B = 3 \). **Fully Factored Form:** \[ 25 a^{2}(a - 3 b) + 9(3 b - a) = (a - 3b)(5a - 3)(5a + 3) \] --- Feel free to ask if you need further explanations or assistance with these problems!