Divide using synthetic division $$ ( x^{2}-x+3 ) \div ( -2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-3&1&-1&3\\& & -3& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{orangered}{15} \end{array} $$The solution is:
$$ \frac{ x^{2}-x+3 }{ x+3 } = \color{blue}{x-4} ~+~ \frac{ \color{red}{ 15 } }{ x+3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-3}&1&-1&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-3&\color{orangered}{ 1 }&-1&3\\& & & \\ \hline &\color{orangered}{1}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&1&-1&3\\& & \color{blue}{-3} & \\ \hline &\color{blue}{1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrr}-3&1&\color{orangered}{ -1 }&3\\& & \color{orangered}{-3} & \\ \hline &1&\color{orangered}{-4}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrr}\color{blue}{-3}&1&-1&3\\& & -3& \color{blue}{12} \\ \hline &1&\color{blue}{-4}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 12 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrr}-3&1&-1&\color{orangered}{ 3 }\\& & -3& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{-4}&\color{orangered}{15} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x-4 } $ with a remainder of $ \color{red}{ 15 } $.