Use synthetic division to divide the polynomials. \[ \left(5-8 x-6 x^{2}-5 x^{4}\right) \div(x+1)=\square \]

We first write the dividend polynomial in standard form with all degrees present. Given \[ 5-8x-6x^2-5x^4, \] we rewrite it as \[ -5x^4+0x^3-6x^2-8x+5. \] Since we are dividing by \(x+1\), the zero is \[ x=-1. \] Now, we list the coefficients: - Coefficient for \(x^4\): \(-5\) - Coefficient for \(x^3\): \(0\) - Coefficient for \(x^2\): \(-6\) - Coefficient for \(x\): \(-8\) - Constant term: \(5\) Next, we set up synthetic division with \(-1\): \[ \begin{array}{r|rrrrr} -1 & -5 & 0 & -6 & -8 & 5 \\ & & & & & \\ \hline & -5 & & & & \end{array} \] 1. **Bring down** the first coefficient: \[ -5. \] 2. **Multiply** \(-5\) by \(-1\) and place the result under the next coefficient: \[ -5 \times (-1)=5. \] Add to the next coefficient: \[ 0+5=5. \] 3. **Multiply** \(5\) by \(-1\): \[ 5 \times (-1)=-5. \] Add to the next coefficient: \[ -6+(-5)=-11. \] 4. **Multiply** \(-11\) by \(-1\): \[ -11 \times (-1)=11. \] Add to the next coefficient: \[ -8+11=3. \] 5. **Multiply** \(3\) by \(-1\): \[ 3 \times (-1)=-3. \] Add to the constant term: \[ 5+(-3)=2. \] The synthetic division table now looks like: \[ \begin{array}{r|rrrrr} -1 & -5 & 0 & -6 & -8 & 5 \\ & & 5 & -5 & 11 & -3\\ \hline & -5 & 5 & -11 & 3 & 2\\ \end{array} \] The bottom row, excluding the final number, gives the coefficients of the quotient polynomial: \[ -5x^3+5x^2-11x+3, \] and the final number, \(2\), is the remainder. Thus, the result of the division is: \[ -5x^3+5x^2-11x+3+\frac{2}{x+1}. \]