Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.

$$ \begin{array}{c|rrr}\color{blue}{-2}&3&5&-2\\& & & \\ \hline &&& \end{array} $$

Step 1 : Bring down the leading coefficient to the bottom row.

$$ \begin{array}{c|rrr}-2&\color{orangered}{ 3 }&5&-2\\& & & \\ \hline &\color{orangered}{3}&& \end{array} $$

Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.

$$ \begin{array}{c|rrr}\color{blue}{-2}&3&5&-2\\& & \color{blue}{-6} & \\ \hline &\color{blue}{3}&& \end{array} $$

Step 3 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -1 } $

$$ \begin{array}{c|rrr}-2&3&\color{orangered}{ 5 }&-2\\& & \color{orangered}{-6} & \\ \hline &3&\color{orangered}{-1}& \end{array} $$

Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 2 } $.

$$ \begin{array}{c|rrr}\color{blue}{-2}&3&5&-2\\& & -6& \color{blue}{2} \\ \hline &3&\color{blue}{-1}& \end{array} $$

Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 2 } = \color{orangered}{ 0 } $

$$ \begin{array}{c|rrr}-2&3&5&\color{orangered}{ -2 }\\& & -6& \color{orangered}{2} \\ \hline &\color{blue}{3}&\color{blue}{-1}&\color{orangered}{0} \end{array} $$

Bottom line represents the quotient $ \color{blue}{ 3x-1 } $ with a remainder of $ \color{red}{ 0 } $.

Synthetic Division Calculator