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

To perform synthetic division, we start by rewriting the coefficients of the polynomial \(4x^3 - 2x^2 + 5x - 2\) as: \(4, -2, 5, -2\). Since we're dividing by \(x - 1\), we use \(1\) for synthetic division. Here are the steps: 1. Write down the coefficients: \(4, -2, 5, -2\). 2. Bring down the leading coefficient (4) below the line. 3. Multiply \(1\) (the value from \(x - 1\)) by \(4\), which gives \(4\), and add it to \(-2\): \(-2 + 4 = 2\). 4. Multiply \(1\) by \(2\) and add it to \(5\): \(5 + 2 = 7\). 5. Multiply \(1\) by \(7\) and add it to \(-2\): \(-2 + 7 = 5\). So the synthetic division yields: \[ \begin{array}{r|rrrr} 1 & 4 & -2 & 5 & -2 \\ & & 4 & 2 & 7 \\ \hline & 4 & 2 & 7 & 5 \\ \end{array} \] The result is \(4x^2 + 2x + 7\) with a remainder of \(5\). Thus, the answer is: \[ \left(4 x^{3}-2 x^{2}+5 x-2\right) \div(x-1) = 4x^2 + 2x + 7 + \frac{5}{x-1} \]