Divide. If the polynomial does not divide evenly, include the remainder as a fraction. \[ \left(-10 v^{3}-54 v^{2}+27 v-56\right) \div(v+6) \]

To divide the polynomial \(-10v^{3} - 54v^{2} + 27v - 56\) by \(v + 6\), we can use **synthetic division**. ### Steps: 1. **Set Up Synthetic Division:** - The divisor is \(v + 6\), so we use \(v = -6\) for synthetic division. - Write down the coefficients of the polynomial: \(-10\), \(-54\), \(27\), \(-56\). 2. **Perform Synthetic Division:** \[ \begin{array}{c|cccc} -6 & -10 & -54 & 27 & -56 \\ & & 60 & -6 & 54 \\ \hline & -10 & 6 & -9 & -2 \\ \end{array} \] - **Step 1:** Bring down the first coefficient \(-10\). - **Step 2:** Multiply \(-10\) by \(-6\) to get \(60\) and add to the next coefficient: \(-54 + 60 = 6\). - **Step 3:** Multiply \(6\) by \(-6\) to get \(-36\) and add to the next coefficient: \(27 - 36 = -9\). - **Step 4:** Multiply \(-9\) by \(-6\) to get \(54\) and add to the last coefficient: \(-56 + 54 = -2\). 3. **Interpret the Results:** - The numbers at the bottom row represent the coefficients of the quotient and the remainder. - The quotient is \(-10v^{2} + 6v - 9\). - The remainder is \(-2\). 4. **Write the Final Answer:** Since the polynomial does not divide evenly, include the remainder as a fraction over the original divisor. \[ \boxed{-10v^{2} + 6v - 9 - \frac{2}{v + 6}} \]