Divide using synthetic division $$ ( x^{3}+3x^{2}-1 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}3&1&3&0&-1\\& & 3& 18& \color{black}{54} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{53} \end{array} $$The solution is:
$$ \dfrac{ x^{3}+3x^{2}-1 }{ x-3 } = \color{blue}{x^{2}+6x+18} ~+~ \dfrac{ \color{red}{ 53 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&3&0&-1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}3&\color{orangered}{ 1 }&3&0&-1\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&3&0&-1\\& & \color{blue}{3} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 3 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrrr}3&1&\color{orangered}{ 3 }&0&-1\\& & \color{orangered}{3} & & \\ \hline &1&\color{orangered}{6}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 6 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&3&0&-1\\& & 3& \color{blue}{18} & \\ \hline &1&\color{blue}{6}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrr}3&1&3&\color{orangered}{ 0 }&-1\\& & 3& \color{orangered}{18} & \\ \hline &1&6&\color{orangered}{18}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 18 } = \color{blue}{ 54 } $.
$$ \begin{array}{c|rrrr}\color{blue}{3}&1&3&0&-1\\& & 3& 18& \color{blue}{54} \\ \hline &1&6&\color{blue}{18}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 54 } = \color{orangered}{ 53 } $
$$ \begin{array}{c|rrrr}3&1&3&0&\color{orangered}{ -1 }\\& & 3& 18& \color{orangered}{54} \\ \hline &\color{blue}{1}&\color{blue}{6}&\color{blue}{18}&\color{orangered}{53} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}+6x+18 } $ with a remainder of $ \color{red}{ 53 } $.