Factor Polynomial 80B5D9

1. **State the problem:** Factor the polynomial $$x^4 - 4x^2 - 5$$ completely over the integers. 2. **Recognize the form:** Notice that the polynomial is a quadratic in terms of $x^2$. Let $y = x^2$, then the expression becomes $$y^2 - 4y - 5$$. 3. **Factor the quadratic in $y$:** We look for two numbers that multiply to $-5$ and add to $-4$. These numbers are $-5$ and $1$. 4. **Write the factorization in $y$:** $$y^2 - 4y - 5 = (y - 5)(y + 1)$$ 5. **Substitute back $y = x^2$:** $$(x^2 - 5)(x^2 + 1)$$ 6. **Check for further factorization:** - $x^2 - 5$ cannot be factored further over the integers because $5$ is not a perfect square. - $x^2 + 1$ cannot be factored over the integers because it has no real roots. 7. **Final factorization:** $$\boxed{(x^2 - 5)(x^2 + 1)}$$