Divide using synthetic division. \[ \left(4 x^{3}-2 x^{2}+5 x-2\right)+(x-1) \] \( \left(4 x^{3}-2 x^{2}+5 x-2\right)+(x-1)=\square \) (Simplify your answer. Do not factor.)

1. Write the divisor in the form \(x-c\). The divisor is \(x-1\), so \(c=1\). 2. List the coefficients of the dividend \(4x^3-2x^2+5x-2\): - Coefficient of \(x^3\): 4 - Coefficient of \(x^2\): \(-2\) - Coefficient of \(x\): 5 - Constant term: \(-2\) 3. Set up synthetic division with \(c=1\): \[ \begin{array}{r|rrrr} 1 & 4 & -2 & 5 & -2 \\[6pt] & & 4 & 2 & 7 \\ \hline & 4 & 2 & 7 & 5 \\ \end{array} \] - Bring down the 4. - Multiply \(4 \times 1=4\) and add to \(-2\) to get \(2\). - Multiply \(2 \times 1=2\) and add to \(5\) to get \(7\). - Multiply \(7 \times 1=7\) and add to \(-2\) to get \(5\). 4. The bottom row shows the coefficients of the quotient and the remainder. The quotient is given by the first three numbers as: \[ 4x^2+2x+7 \] and the remainder is 5. 5. Write the result in division form: \[ \frac{4x^3-2x^2+5x-2}{x-1} = 4x^2+2x+7+\frac{5}{x-1}. \] Thus, the simplified answer is: \[ 4x^2+2x+7+\frac{5}{x-1}. \]