Divide using synthetic division $$ ( -2x^{2}+6x+1 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}3&-2&6&1\\& & -6& \color{black}{0} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{orangered}{1} \end{array} $$The solution is:
$$ \frac{ -2x^{2}+6x+1 }{ x-3 } = \color{blue}{-2x} ~+~ \frac{ \color{red}{ 1 } }{ x-3 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&6&1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}3&\color{orangered}{ -2 }&6&1\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&6&1\\& & \color{blue}{-6} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrr}3&-2&\color{orangered}{ 6 }&1\\& & \color{orangered}{-6} & \\ \hline &-2&\color{orangered}{0}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{3}&-2&6&1\\& & -6& \color{blue}{0} \\ \hline &-2&\color{blue}{0}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrr}3&-2&6&\color{orangered}{ 1 }\\& & -6& \color{orangered}{0} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{orangered}{1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x } $ with a remainder of $ \color{red}{ 1 } $.