1. **State the problem:** Simplify the expression $$1 - \frac{a^2 - 2}{a^2 - 16} + \frac{4 - a}{a^3 - 4a^2 - 16a + 64}$$. 2. **Identify and factor denominators and numerators where possible:** - Factor the denominator $a^2 - 16$ as a difference of squares: $$a^2 - 16 = (a - 4)(a + 4)$$ - Factor the cubic denominator $a^3 - 4a^2 - 16a + 64$ by grouping: $$a^3 - 4a^2 - 16a + 64 = (a^3 - 4a^2) - (16a - 64) = a^2(a - 4) - 16(a - 4) = (a - 4)(a^2 - 16)$$ - Since $a^2 - 16 = (a - 4)(a + 4)$, the cubic denominator factors as: $$a^3 - 4a^2 - 16a + 64 = (a - 4)^2 (a + 4)$$ 3. **Rewrite the expression with factored denominators:** $$1 - \frac{a^2 - 2}{(a - 4)(a + 4)} + \frac{4 - a}{(a - 4)^2 (a + 4)}$$ 4. **Rewrite numerator $4 - a$ as $-(a - 4)$ to simplify:** $$\frac{4 - a}{(a - 4)^2 (a + 4)} = \frac{-(a - 4)}{(a - 4)^2 (a + 4)} = -\frac{1}{(a - 4)(a + 4)}$$ 5. **Substitute back into the expression:** $$1 - \frac{a^2 - 2}{(a - 4)(a + 4)} - \frac{1}{(a - 4)(a + 4)}$$ 6. **Combine the two fractions with common denominator:** $$1 - \frac{a^2 - 2 + 1}{(a - 4)(a + 4)} = 1 - \frac{a^2 - 1}{(a - 4)(a + 4)}$$ 7. **Factor numerator $a^2 - 1$ as difference of squares:** $$a^2 - 1 = (a - 1)(a + 1)$$ 8. **Rewrite the expression:** $$1 - \frac{(a - 1)(a + 1)}{(a - 4)(a + 4)}$$ 9. **Write 1 as a fraction with the same denominator:** $$\frac{(a - 4)(a + 4)}{(a - 4)(a + 4)} - \frac{(a - 1)(a + 1)}{(a - 4)(a + 4)} = \frac{(a - 4)(a + 4) - (a - 1)(a + 1)}{(a - 4)(a + 4)}$$ 10. **Expand the numerators:** - $$(a - 4)(a + 4) = a^2 - 16$$ - $$(a - 1)(a + 1) = a^2 - 1$$ 11. **Subtract the numerators:** $$a^2 - 16 - (a^2 - 1) = a^2 - 16 - a^2 + 1 = -15$$ 12. **Final simplified expression:** $$\frac{-15}{(a - 4)(a + 4)} = \frac{-15}{a^2 - 16}$$ **Answer:** $$\boxed{\frac{-15}{a^2 - 16}}$$