Use synthetic division to find the result when \( 4 x^{4}+25 x^{3}+2 x^{2}-25 x-8 \) is divided by \( x+6 \). If there is a remainder, express the result in the form \( q(x)+\frac{r(x)}{b(x)} \)

To perform synthetic division of the polynomial \( 4x^4 + 25x^3 + 2x^2 - 25x - 8 \) by \( x + 6 \), we first note that we can use \( -6 \) as the value for synthetic division. We will set up the synthetic division as follows: 1. Write down the coefficients of the polynomial: - The coefficients are \( 4, 25, 2, -25, -8 \). 2. Set up the synthetic division with \( -6 \): \[ \begin{array}{r|rrrrr} -6 & 4 & 25 & 2 & -25 & -8 \\ & & -24 & -6 & 24 & 6 \\ \hline & 4 & 1 & -4 & -1 & -2 \\ \end{array} \] 3. Perform the synthetic division: - Bring down the leading coefficient \( 4 \). - Multiply \( 4 \) by \( -6 \) to get \( -24 \) and add it to \( 25 \) to get \( 1 \). - Multiply \( 1 \) by \( -6 \) to get \( -6 \) and add it to \( 2 \) to get \( -4 \). - Multiply \( -4 \) by \( -6 \) to get \( 24 \) and add it to \( -25 \) to get \( -1 \). - Multiply \( -1 \) by \( -6 \) to get \( 6 \) and add it to \( -8 \) to get \( -2 \). 4. The result of the synthetic division gives us the coefficients of the quotient and the remainder: - The quotient is \( 4x^3 + 1x^2 - 4x - 1 \). - The remainder is \( -2 \). Thus, we can express the result of the division as: \[ q(x) + \frac{r(x)}{b(x)} = 4x^3 + x^2 - 4x - 1 + \frac{-2}{x + 6} \] Therefore, the final result is: \[ 4x^3 + x^2 - 4x - 1 - \frac{2}{x + 6} \]