Divide using Synthetic division. 1) \( \quad\left(-8 x^{3}+40 x^{2}-37 x+30\right) \div(x-4) \)

To perform synthetic division of the polynomial \(-8x^3 + 40x^2 - 37x + 30\) by \(x - 4\), we will use the root of the divisor, which is \(4\). Here are the steps for synthetic division: 1. Write down the coefficients of the polynomial: \[ -8, \quad 40, \quad -37, \quad 30 \] 2. Set up the synthetic division by placing \(4\) to the left and the coefficients to the right: \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & & & \\ \end{array} \] 3. Bring down the leading coefficient \(-8\): \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & & & \\ & -8 & & & \\ \end{array} \] 4. Multiply \(4\) by \(-8\) and write the result under the next coefficient: \[ 4 \times -8 = -32 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & & \\ & -8 & & & \\ \end{array} \] 5. Add the second column: \[ 40 + (-32) = 8 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & & \\ & -8 & 8 & & \\ \end{array} \] 6. Multiply \(4\) by \(8\) and write the result under the next coefficient: \[ 4 \times 8 = 32 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & 32 & \\ & -8 & 8 & & \\ \end{array} \] 7. Add the third column: \[ -37 + 32 = -5 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & 32 & \\ & -8 & 8 & -5 & \\ \end{array} \] 8. Multiply \(4\) by \(-5\) and write the result under the last coefficient: \[ 4 \times -5 = -20 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & 32 & -20 \\ & -8 & 8 & -5 & \\ \end{array} \] 9. Add the last column: \[ 30 + (-20) = 10 \] \[ \begin{array}{r|rrrr} 4 & -8 & 40 & -37 & 30 \\ & & -32 & 32 & -20 \\ & -8 & 8 & -5 & 10 \\ \end{array} \] The result of the synthetic division is: - The coefficients of the quotient polynomial are \(-8, 8, -5\), which corresponds to \(-8x^2 + 8x - 5\). - The remainder is \(10\). Thus, we can write the result of the division as: \[ -8x^2 + 8x - 5 + \frac{10}{x - 4} \] So, the final answer is: \[ \boxed{-8x^2 + 8x - 5 + \frac{10}{x - 4}} \]