1. **State the problem:** We need to divide the polynomial $$x^4 - 8x^3 + (5a - 1)x^2 + 6x - 3a - 6$$ by $$x^2 - 1$$. 2. **Recall the formula and rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by that result, subtract, and repeat until the degree of the remainder is less than the divisor. 3. **Perform the division:** Divide $$x^4$$ by $$x^2$$ to get $$x^2$$. Multiply $$x^2$$ by $$x^2 - 1$$ to get $$x^4 - x^2$$. Subtract: $$\left(x^4 - 8x^3 + (5a - 1)x^2 + 6x - 3a - 6\right) - \left(x^4 - x^2\right) = -8x^3 + (5a - 1 + 1)x^2 + 6x - 3a - 6 = -8x^3 + 5a x^2 + 6x - 3a - 6$$. 4. Divide $$-8x^3$$ by $$x^2$$ to get $$-8x$$. Multiply $$-8x$$ by $$x^2 - 1$$ to get $$-8x^3 + 8x$$. Subtract: $$\left(-8x^3 + 5a x^2 + 6x - 3a - 6\right) - \left(-8x^3 + 8x\right) = 5a x^2 + (6x - 8x) - 3a - 6 = 5a x^2 - 2x - 3a - 6$$. 5. Divide $$5a x^2$$ by $$x^2$$ to get $$5a$$. Multiply $$5a$$ by $$x^2 - 1$$ to get $$5a x^2 - 5a$$. Subtract: $$\left(5a x^2 - 2x - 3a - 6\right) - \left(5a x^2 - 5a\right) = -2x + (-3a - 6 + 5a) = -2x + 2a - 6$$. 6. The remainder is $$-2x + 2a - 6$$, which has degree 1, less than degree 2 of the divisor, so division stops. 7. **Final answer:** $$\frac{x^4 - 8x^3 + (5a - 1)x^2 + 6x - 3a - 6}{x^2 - 1} = x^2 - 8x + 5a + \frac{-2x + 2a - 6}{x^2 - 1}$$