Factor completely. If a polynomial cannot be factored, say it is prime. \( x^{4}-3 x^{2}-10 \)

To factor the polynomial \( x^{4} - 3x^{2} - 10 \) completely, follow these steps: 1. **Treat it as a Quadratic in \( x^2 \):** Let \( y = x^2 \). Then the polynomial becomes: \[ y^2 - 3y - 10 \] 2. **Factor the Quadratic:** We need to find two numbers that multiply to \(-10\) and add to \(-3\). These numbers are \(-5\) and \(2\). \[ y^2 - 3y - 10 = (y - 5)(y + 2) \] 3. **Substitute Back \( y = x^2 \):** \[ (y - 5)(y + 2) = (x^2 - 5)(x^2 + 2) \] 4. **Check for Further Factoring:** - \( x^2 - 5 \) can be factored over the real numbers as \( (x - \sqrt{5})(x + \sqrt{5}) \), but typically "factoring completely" refers to factoring over the integers unless otherwise specified. - \( x^2 + 2 \) does not factor further over the real numbers. Therefore, the completely factored form over the integers is: \[ (x^2 - 5)(x^2 + 2) \] **Answer:** After factoring, the polynomial is (x² – 5) multiplied by (x² + 2). Thus, (x² − 5)(x² + 2)