1. **State the problem:** We need to divide the polynomial $$x^3 - x^2 - 17x - 15$$ by $$x - 5$$ using synthetic division and find the quotient. 2. **Recall synthetic division:** When dividing by $$x - c$$, we use synthetic division with the value $$c$$. 3. **Set up synthetic division:** Here, $$c = 5$$. Write the coefficients of the dividend polynomial: $$1$$ (for $$x^3$$), $$-1$$ (for $$x^2$$), $$-17$$ (for $$x$$), and $$-15$$ (constant). 4. **Perform synthetic division:** Start with coefficients: $$1, -1, -17, -15$$ Bring down the first coefficient: $$1$$ Multiply by $$c=5$$: $$1 \times 5 = 5$$ Add to next coefficient: $$-1 + 5 = 4$$ Multiply by $$5$$: $$4 \times 5 = 20$$ Add to next coefficient: $$-17 + 20 = 3$$ Multiply by $$5$$: $$3 \times 5 = 15$$ Add to next coefficient: $$-15 + 15 = 0$$ 5. **Write the quotient:** The numbers $$1, 4, 3$$ correspond to coefficients of the quotient polynomial of degree one less than the dividend, so the quotient is: $$x^2 + 4x + 3$$ 6. **Interpret remainder:** The remainder is $$0$$, so division is exact. **Final answer:** $$x^2 + 4x + 3$$