Factor Polynomial 18Cc6D

1. **State the problem:** Factor the polynomial $$x^4 - 2x^2 - 15$$ completely over the integers. 2. **Identify the structure:** Notice that the polynomial is in terms of $x^4$, $x^2$, and a constant. We can treat it as a quadratic in $x^2$ by letting $y = x^2$. 3. **Rewrite the polynomial:** $$x^4 - 2x^2 - 15 = y^2 - 2y - 15$$ 4. **Factor the quadratic in $y$:** We look for two numbers that multiply to $-15$ and add to $-2$. These numbers are $-5$ and $3$. 5. **Factorization:** $$y^2 - 2y - 15 = (y - 5)(y + 3)$$ 6. **Substitute back $y = x^2$:** $$(x^2 - 5)(x^2 + 3)$$ 7. **Check for further factorization:** - $x^2 - 5$ can be factored over the integers only if $5$ is a perfect square, which it is not. - $x^2 + 3$ cannot be factored over the integers because it has no real roots. 8. **Final factorization:** $$\boxed{(x^2 - 5)(x^2 + 3)}$$