Divide using synthetic division $$ ( -3x^{2}+5x-2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}2&-3&5&-2\\& & -6& \color{black}{-2} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{-4} \end{array} $$The solution is:
$$ \frac{ -3x^{2}+5x-2 }{ x-2 } = \color{blue}{-3x-1} \color{red}{~-~} \frac{ \color{red}{ 4 } }{ x-2 } $$Explanation
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}{ \left( -3 \right) } = \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}{ \left( -2 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrr}2&-3&5&\color{orangered}{ -2 }\\& & -6& \color{orangered}{-2} \\ \hline &\color{blue}{-3}&\color{blue}{-1}&\color{orangered}{-4} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x-1 } $ with a remainder of $ \color{red}{ -4 } $.