1. **State the problem:** We need to divide the polynomial $$3x^4 - 5x^3 + 0x^2 + 0x + 7$$ by $$x + 1$$ using synthetic division. 2. **Set up synthetic division:** Since we divide by $$x + 1$$, we use the root $$-1$$ for synthetic division. 3. **Write coefficients:** The coefficients of the dividend polynomial are $$3, -5, 0, 0, 7$$. 4. **Perform synthetic division:** - Bring down the first coefficient: $$3$$. - Multiply $$3 \times (-1) = -3$$, add to next coefficient: $$-5 + (-3) = -8$$. - Multiply $$-8 \times (-1) = 8$$, add to next coefficient: $$0 + 8 = 8$$. - Multiply $$8 \times (-1) = -8$$, add to next coefficient: $$0 + (-8) = -8$$. - Multiply $$-8 \times (-1) = 8$$, add to next coefficient: $$7 + 8 = 15$$. 5. **Write the quotient and remainder:** The quotient polynomial has coefficients $$3, -8, 8, -8$$ corresponding to $$3x^3 - 8x^2 + 8x - 8$$ and the remainder is $$15$$. 6. **Final answer:** $$\frac{3x^4 - 5x^3 + 7}{x + 1} = 3x^3 - 8x^2 + 8x - 8 + \frac{15}{x + 1}$$