1. **State the problem:** We need to divide the polynomial $$x^4 + 4x^3 - 24x^2 + 8x + 20$$ by the polynomial divisor $$x^2 - 2x - 3$$ and express the result in the form $$q(x) + \frac{r(x)}{x^2 - 2x - 3}$$ where $$q(x)$$ is the quotient and $$r(x)$$ is the remainder. 2. **Recall the division algorithm for polynomials:** When dividing a polynomial $$f(x)$$ by a polynomial $$d(x)$$, there exist unique polynomials $$q(x)$$ and $$r(x)$$ such that: $$ f(x) = d(x) \cdot q(x) + r(x) $$ where the degree of $$r(x)$$ is less than the degree of $$d(x)$$. 3. **Perform polynomial long division:** - Divide the leading term of the dividend $$x^4$$ by the leading term of the divisor $$x^2$$ to get $$x^2$$. - Multiply the divisor by $$x^2$$: $$x^2(x^2 - 2x - 3) = x^4 - 2x^3 - 3x^2$$. - Subtract this from the dividend: $$ (x^4 + 4x^3 - 24x^2 + 8x + 20) - (x^4 - 2x^3 - 3x^2) = 6x^3 - 21x^2 + 8x + 20 $$. 4. **Repeat the division with the new polynomial:** - Divide the leading term $$6x^3$$ by $$x^2$$ to get $$6x$$. - Multiply the divisor by $$6x$$: $$6x(x^2 - 2x - 3) = 6x^3 - 12x^2 - 18x$$. - Subtract this from the current polynomial: $$ (6x^3 - 21x^2 + 8x + 20) - (6x^3 - 12x^2 - 18x) = -9x^2 + 26x + 20 $$. 5. **Continue the division:** - Divide the leading term $$-9x^2$$ by $$x^2$$ to get $$-9$$. - Multiply the divisor by $$-9$$: $$-9(x^2 - 2x - 3) = -9x^2 + 18x + 27$$. - Subtract this from the current polynomial: $$ (-9x^2 + 26x + 20) - (-9x^2 + 18x + 27) = 8x - 7 $$. 6. **Determine the remainder:** - The remainder is $$8x - 7$$, which has degree 1, less than the degree 2 of the divisor. 7. **Write the final answer:** $$$ \frac{x^4 + 4x^3 - 24x^2 + 8x + 20}{x^2 - 2x - 3} = x^2 + 6x - 9 + \frac{8x - 7}{x^2 - 2x - 3} $$$ This matches the division result format requested.