1. **State the problem:** We need to divide the polynomial $$x^{53} - 12x^{40} - 3x^{27} - 5x^{21} + x^{10} - 3$$ by $$x + 1$$. 2. **Formula and rule:** Polynomial division can be done using synthetic division or long division. Here, we use synthetic division since the divisor is linear of the form $$x - r$$, where $$r = -1$$. 3. **Set up synthetic division:** The coefficients of the dividend polynomial are: $$1, 0, 0, \ldots, 0, -12, 0, \ldots, 0, -3, 0, \ldots, 0, -5, 0, \ldots, 0, 1, 0, \ldots, 0, -3$$ with zeros for all missing powers. We align coefficients for powers 53 down to 0. 4. **Perform synthetic division:** Using $$r = -1$$, bring down the leading coefficient 1, multiply by $$-1$$, add to next coefficient, continue this process. 5. **Result:** The quotient polynomial is $$x^{52} - x^{51} + x^{50} - x^{49} + \cdots - x + 1 - \frac{4}{x+1}$$ 6. **Check remainder:** The remainder is $$-4$$, so the division is $$\frac{x^{53} - 12x^{40} - 3x^{27} - 5x^{21} + x^{10} - 3}{x+1} = Q(x) - \frac{4}{x+1}$$ where $$Q(x)$$ is the quotient polynomial. **Note:** The quotient is a polynomial of degree 52 with alternating signs due to the synthetic division with $$r = -1$$. **Final answer:** $$\frac{x^{53} - 12x^{40} - 3x^{27} - 5x^{21} + x^{10} - 3}{x+1} = x^{52} - x^{51} + x^{50} - x^{49} + \cdots + 1 - \frac{4}{x+1}$$