Divide using synthetic division $$ ( -3x^{2}+4x-8 ) \div ( 2x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-3&4&-8\\& & -9& \color{black}{-15} \\ \hline &\color{blue}{-3}&\color{blue}{-5}&\color{orangered}{-23} \end{array} $$The solution is:
$$ \frac{ -3x^{2}+4x-8 }{ x-3 } = \color{blue}{-3x-5} \color{red}{~-~} \frac{ \color{red}{ 23 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{3}&-3&4&-8\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -3 }&4&-8\\& & & \\ \hline &\color{orangered}{-3}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-3&4&-8\\& & \color{blue}{-9} & \\ \hline &\color{blue}{-3}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrr}3&-3&\color{orangered}{ 4 }&-8\\& & \color{orangered}{-9} & \\ \hline &-3&\color{orangered}{-5}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ -15 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-3&4&-8\\& & -9& \color{blue}{-15} \\ \hline &-3&\color{blue}{-5}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -15 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrr}3&-3&4&\color{orangered}{ -8 }\\& & -9& \color{orangered}{-15} \\ \hline &\color{blue}{-3}&\color{blue}{-5}&\color{orangered}{-23} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x-5 } $ with a remainder of $ \color{red}{ -23 } $.