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

To divide the polynomial \(5-8x-6x^2-5x^4\) by \(x+1\) using synthetic division, we start with the root of \(x + 1 = 0\) which is \(x = -1\). We arrange the coefficients of the polynomial in descending order, filling in any missing coefficients: - Coefficients: \(-5\) (for \(x^4\)), \(0\) (for \(x^3\)), \(-6\) (for \(x^2\)), \(-8\) (for \(x^1\)), and \(5\) (constant term). Now we perform synthetic division: ``` -1 | -5 0 -6 -8 5 | 5 -5 11 -5 ------------------------- -5 5 -11 3 0 ``` The bottom row provides the coefficients of the quotient polynomial, which is \(-5x^3 + 5x^2 - 11x + 3\), and the remainder is \(0\). Thus, the result of the division is: \[ 5 - 8x - 6x^2 - 5x^4 = (x + 1)(-5x^3 + 5x^2 - 11x + 3) \] The answer box can be filled with the quotient: \[ -5x^3 + 5x^2 - 11x + 3 \]