Divide using synthetic division $$ ( 3x^{2}+5x-2 ) \div ( x-2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}2&3&5&-2\\& & 6& \color{black}{22} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{orangered}{20} \end{array} $$The solution is:
$$ \dfrac{ 3x^{2}+5x-2 }{ x-2 } = \color{blue}{3x+11} ~+~ \dfrac{ \color{red}{ 20 } }{ x-2 } $$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}{ 6 } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrr}2&3&\color{orangered}{ 5 }&-2\\& & \color{orangered}{6} & \\ \hline &3&\color{orangered}{11}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 11 } = \color{blue}{ 22 } $.
$$ \begin{array}{c|rrr}\color{blue}{2}&3&5&-2\\& & 6& \color{blue}{22} \\ \hline &3&\color{blue}{11}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 22 } = \color{orangered}{ 20 } $
$$ \begin{array}{c|rrr}2&3&5&\color{orangered}{ -2 }\\& & 6& \color{orangered}{22} \\ \hline &\color{blue}{3}&\color{blue}{11}&\color{orangered}{20} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x+11 } $ with a remainder of $ \color{red}{ 20 } $.