1. **State the problem:** Simplify the expression $$\frac{x^2 - 1}{x^3 + 8} \times \frac{x^9 - x^8 - 6x^7}{x^5 + x^4} \div \frac{x^3 - 4x^2 + x + 6}{x^4 - 16}$$ 2. **Rewrite division as multiplication by reciprocal:** $$\frac{x^2 - 1}{x^3 + 8} \times \frac{x^9 - x^8 - 6x^7}{x^5 + x^4} \times \frac{x^4 - 16}{x^3 - 4x^2 + x + 6}$$ 3. **Factor all polynomials:** - $x^2 - 1 = (x - 1)(x + 1)$ (difference of squares) - $x^3 + 8 = (x + 2)(x^2 - 2x + 4)$ (sum of cubes) - $x^9 - x^8 - 6x^7 = x^7(x^2 - x - 6) = x^7(x - 3)(x + 2)$ - $x^5 + x^4 = x^4(x + 1)$ - $x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)$ - $x^3 - 4x^2 + x + 6$ factor by grouping: $$x^3 - 4x^2 + x + 6 = (x^3 - 4x^2) + (x + 6) = x^2(x - 4) + 1(x + 6)$$ Try factoring further: Test roots: $x=2$: $$2^3 - 4(2)^2 + 2 + 6 = 8 - 16 + 2 + 6 = 0$$ So $(x - 2)$ is a factor. Divide polynomial by $(x - 2)$: $$\frac{x^3 - 4x^2 + x + 6}{x - 2} = x^2 - 2x - 3$$ Factor quadratic: $$x^2 - 2x - 3 = (x - 3)(x + 1)$$ So, $$x^3 - 4x^2 + x + 6 = (x - 2)(x - 3)(x + 1)$$ 4. **Rewrite the expression with factors:** $$\frac{(x - 1)(x + 1)}{(x + 2)(x^2 - 2x + 4)} \times \frac{x^7 (x - 3)(x + 2)}{x^4 (x + 1)} \times \frac{(x - 2)(x + 2)(x^2 + 4)}{(x - 2)(x - 3)(x + 1)}$$ 5. **Combine all into a single fraction:** $$\frac{(x - 1)(x + 1) \times x^7 (x - 3)(x + 2) \times (x - 2)(x + 2)(x^2 + 4)}{(x + 2)(x^2 - 2x + 4) \times x^4 (x + 1) \times (x - 2)(x - 3)(x + 1)}$$ 6. **Cancel common factors:** - Cancel $(x + 2)$ numerator and denominator: $$\frac{\cancel{(x + 2)}}{\cancel{(x + 2)}}$$ - Cancel $(x - 3)$ numerator and denominator: $$\frac{\cancel{(x - 3)}}{\cancel{(x - 3)}}$$ - Cancel $(x - 2)$ numerator and denominator: $$\frac{\cancel{(x - 2)}}{\cancel{(x - 2)}}$$ - Cancel $(x + 1)$ numerator and denominator: Numerator has $(x + 1)$ once, denominator has $(x + 1)$ twice, so cancel one: $$\frac{(x + 1)}{(x + 1)(x + 1)} = \frac{\cancel{(x + 1)}}{\cancel{(x + 1)}(x + 1)} = \frac{1}{x + 1}$$ 7. **Simplify powers of $x$:** $$\frac{x^7}{x^4} = x^{7 - 4} = x^3$$ 8. **Write the simplified expression:** $$\frac{(x - 1) \times x^3 \times (x + 2) \times (x^2 + 4)}{(x^2 - 2x + 4) \times (x + 1)}$$ 9. **Final answer:** $$\boxed{\frac{x^3 (x - 1)(x + 2)(x^2 + 4)}{(x^2 - 2x + 4)(x + 1)}}$$ This is the fully simplified form of the original expression.