Choose the simplified form of the following expression: \( (4 x-7)^{2}-(x+5)^{2} \) \( 15 x^{2}+66 x-24 \) \( 17 x^{2}-46 x+24 \) \( 17 x^{2}-23 x+24 \) \( 15 x^{2}-66 x+24 \)

To simplify the expression \( (4x - 7)^{2} - (x + 5)^{2} \), we can apply the difference of squares formula, which states that \( a^{2} - b^{2} = (a - b)(a + b) \). Letting \( a = 4x - 7 \) and \( b = x + 5 \), we find: 1. **Calculate \( a - b \)**: \[ a - b = (4x - 7) - (x + 5) = 4x - 7 - x - 5 = 3x - 12 \] 2. **Calculate \( a + b \)**: \[ a + b = (4x - 7) + (x + 5) = 4x - 7 + x + 5 = 5x - 2 \] So the expression becomes: \[ (3x - 12)(5x - 2) \] Now we will expand this product: \[ = 3x(5x) + 3x(-2) - 12(5x) - 12(-2) = 15x^{2} - 6x - 60x + 24 = 15x^{2} - 66x + 24 \] Thus, the simplified form of the expression \( (4x - 7)^{2} - (x + 5)^{2} \) is: \[ \boxed{15x^{2} - 66x + 24} \] So the correct answer among the options provided is \( 15 x^{2} - 66 x + 24 \).