Use synthetic division to divide. (x^3+14x^2+8x-9)/(x+5) (x^3+14x^2+8x-9)/(x+5)=square (Simplify your answer. Do not factor.)

To use synthetic division to divide \(\frac{x^3 + 14x^2 + 8x - 9}{x + 5}\), we first need to rewrite the divisor \(x + 5\) in the form \(x - c\). In this case, \(c = -5\).<br /><br />The coefficients of the polynomial \(x^3 + 14x^2 + 8x - 9\) are \(1, 14, 8, -9\).<br /><br />Now, we set up the synthetic division:<br /><br />\[<br />\begin{array}{r|rrrr}<br />-5 & 1 & 14 & 8 & -9 \\<br /> & & -5 & -45 & 215 \\<br />\hline<br /> & 1 & 9 & -37 & 206 \\<br />\end{array}<br />\]<br /><br />Here's the step-by-step process:<br /><br />1. Bring down the first coefficient (1).<br />2. Multiply -5 by the value just written below the line (1), which gives -5. Write this under the next coefficient (14).<br />3. Add the values in the second column: \(14 + (-5) = 9\).<br />4. Multiply -5 by the new value (9), which gives -45. Write this under the next coefficient (8).<br />5. Add the values in the third column: \(8 + (-45) = -37\).<br />6. Multiply -5 by the new value (-37), which gives 185. Write this under the next coefficient (-9).<br />7. Add the values in the fourth column: \(-9 + 185 = 176\).<br /><br />The final row gives the coefficients of the quotient polynomial and the remainder. The quotient polynomial is \(x^2 + 9x - 37\) and the remainder is 176.<br /><br />So, the result of the division is:<br /><br />\[<br />\frac{x^3 + 14x^2 + 8x - 9}{x + 5} = x^2 + 9x - 37 + \frac{176}{x + 5}<br />\]<br /><br />In simplified form, without factoring the remainder, the answer is:<br /><br />\[<br />\boxed{x^2 + 9x - 37 + \frac{176}{x + 5}}<br />\]