1. **State the problem:** Simplify the expression $$\frac{x^4 - 9x^2}{x^2 - 1 + 5}$$. 2. **Rewrite the denominator:** Combine like terms in the denominator: $$x^2 - 1 + 5 = x^2 + 4$$ 3. **Factor the numerator:** Factor out the common term $x^2$: $$x^4 - 9x^2 = x^2(x^2 - 9)$$ 4. **Recognize difference of squares:** The term $x^2 - 9$ is a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ 5. **Rewrite the numerator with factors:** $$x^2(x - 3)(x + 3)$$ 6. **Write the full expression with factors:** $$\frac{x^2(x - 3)(x + 3)}{x^2 + 4}$$ 7. **Check for common factors:** The denominator $x^2 + 4$ cannot be factored further over the reals and shares no common factors with the numerator. 8. **Final simplified form:** $$\frac{x^2(x - 3)(x + 3)}{x^2 + 4}$$ This is the simplified form since no further cancellation is possible. **Answer:** $$\frac{x^2(x - 3)(x + 3)}{x^2 + 4}$$