Factor completely \( \begin{array}{l}x^{3}+x^{2}-25 x-25 \\ (x-1)(x+5)(x-5) \\ (x+5)(x-1)(x+5) \\ (x-5)(x+5)\end{array} \)

To factor the given polynomial \(x^{3} + x^{2} - 25x - 25\) completely, start by looking for common factors or using polynomial long division. Factoring by grouping often works well here. Notice that we can group the terms: \[ (x^{3} + x^{2}) + (-25x - 25) = x^{2}(x + 1) - 25(x + 1) \] Now we can factor out the common term \((x + 1)\): \[ (x + 1)(x^{2} - 25) \] The expression \(x^{2} - 25\) is a difference of squares, which can be factored further: \[ (x + 1)(x - 5)(x + 5) \] So the complete factorization of the polynomial is: \[ (x + 1)(x - 5)(x + 5) \] Now let's look at the choices given. The correct answer corresponds to the factorization obtained: 1. \( (x + 5)(x - 1)(x + 5) \) is incorrect since it repeats \(x + 5\) and does not represent the original polynomial. 2. \( (x - 1)(x + 5)(x - 5) \) is also incorrect as it misplaces the correct factors. 3. \( (x - 5)(x + 5) \) is just the difference of squares portion without the \(x + 1\). Thus, none of the provided factorization forms are correct for the original polynomial. The correct complete factorization is: \[ (x + 1)(x - 5)(x + 5) \]