Divide using synthetic division $$ ( -2x^{2}+9x+1 ) \div ( x-6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}6&-2&9&1\\& & -12& \color{black}{-18} \\ \hline &\color{blue}{-2}&\color{blue}{-3}&\color{orangered}{-17} \end{array} $$The solution is:
$$ \frac{ -2x^{2}+9x+1 }{ x-6 } = \color{blue}{-2x-3} \color{red}{~-~} \frac{ \color{red}{ 17 } }{ x-6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -6 = 0 $ ( $ x = \color{blue}{ 6 } $ ) at the left.
$$ \begin{array}{c|rrr}\color{blue}{6}&-2&9&1\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}6&\color{orangered}{ -2 }&9&1\\& & & \\ \hline &\color{orangered}{-2}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-2&9&1\\& & \color{blue}{-12} & \\ \hline &\color{blue}{-2}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrr}6&-2&\color{orangered}{ 9 }&1\\& & \color{orangered}{-12} & \\ \hline &-2&\color{orangered}{-3}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 6 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrr}\color{blue}{6}&-2&9&1\\& & -12& \color{blue}{-18} \\ \hline &-2&\color{blue}{-3}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrr}6&-2&9&\color{orangered}{ 1 }\\& & -12& \color{orangered}{-18} \\ \hline &\color{blue}{-2}&\color{blue}{-3}&\color{orangered}{-17} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x-3 } $ with a remainder of $ \color{red}{ -17 } $.