1. **Stating the problem:** Multiply the polynomials $$ (x^4 - 8x^3 + (5a - 1)x^2 + 8x - 3a - 6) \cdot (x^2 - 1) $$. 2. **Formula and rules:** To multiply two polynomials, distribute each term of the first polynomial by each term of the second polynomial and then combine like terms. 3. **Step-by-step multiplication:** Multiply each term in the first polynomial by $$x^2$$: $$x^4 \cdot x^2 = x^6$$ $$-8x^3 \cdot x^2 = -8x^5$$ $$(5a - 1)x^2 \cdot x^2 = (5a - 1)x^4$$ $$8x \cdot x^2 = 8x^3$$ $$(-3a - 6) \cdot x^2 = (-3a - 6)x^2$$ Multiply each term in the first polynomial by $$-1$$: $$x^4 \cdot (-1) = -x^4$$ $$-8x^3 \cdot (-1) = +8x^3$$ $$(5a - 1)x^2 \cdot (-1) = -(5a - 1)x^2 = -5ax^2 + x^2$$ $$8x \cdot (-1) = -8x$$ $$(-3a - 6) \cdot (-1) = +3a + 6$$ 4. **Combine all terms:** $$x^6 - 8x^5 + (5a - 1)x^4 + 8x^3 + (-3a - 6)x^2 - x^4 + 8x^3 - 5ax^2 + x^2 - 8x + 3a + 6$$ 5. **Group like terms:** - For $$x^6$$: $$x^6$$ - For $$x^5$$: $$-8x^5$$ - For $$x^4$$: $$(5a - 1)x^4 - x^4 = (5a - 2)x^4$$ - For $$x^3$$: $$8x^3 + 8x^3 = 16x^3$$ - For $$x^2$$: $$(-3a - 6)x^2 - 5ax^2 + x^2 = (-3a - 6 - 5a + 1)x^2 = (-8a - 5)x^2$$ - For $$x$$: $$-8x$$ - Constants: $$3a + 6$$ 6. **Final simplified expression:** $$\boxed{x^6 - 8x^5 + (5a - 2)x^4 + 16x^3 + (-8a - 5)x^2 - 8x + 3a + 6}$$