Use synthetic division to divide. (2x^3-6x+8x^2-24)/(x+4) square xneq -4

To use synthetic division to divide \( \frac{2x^3 + 8x^2 - 6x - 24}{x + 4} \), we first rewrite the divisor \( x + 4 \) as \( x - (-4) \). Then, we set up the synthetic division as follows:<br /><br />1. Write down the coefficients of the polynomial: \( 2, 8, -6, -24 \).<br />2. Use the root \( -4 \) for synthetic division.<br /><br />\[<br />\begin{array}{r|rrrr}<br />-4 & 2 & 8 & -6 & -24 \\<br /> & & -8 & 0 & 24 \\<br />\hline<br /> & 2 & 0 & -6 & 0 \\<br />\end{array}<br />\]<br /><br />The steps are:<br />- Bring down the first coefficient (2).<br />- Multiply -4 by the value just written below the line and write the result under the next coefficient.<br />- Add the column of numbers and write the result below the line.<br />- Repeat the process for each column.<br /><br />The quotient is given by the numbers on the bottom row, excluding the remainder. Therefore, the quotient is \( 2x^2 - 6 \).<br /><br />So, the answer is:<br />\[ \frac{2x^3 + 8x^2 - 6x - 24}{x + 4} = 2x^2 - 6 \]