Divide using synthetic division $$ ( x^{4}-12x^{3}-69x+192 ) \div ( x-3 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}3&1&-12&0&-69&192\\& & 3& -27& -81& \color{black}{-450} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-27}&\color{blue}{-150}&\color{orangered}{-258} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-12x^{3}-69x+192 }{ x-3 } = \color{blue}{x^{3}-9x^{2}-27x-150} \color{red}{~-~} \dfrac{ \color{red}{ 258 } }{ x-3 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&0&-69&192\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}3&\color{orangered}{ 1 }&-12&0&-69&192\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&0&-69&192\\& & \color{blue}{3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 3 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}3&1&\color{orangered}{ -12 }&0&-69&192\\& & \color{orangered}{3} & & & \\ \hline &1&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&0&-69&192\\& & 3& \color{blue}{-27} & & \\ \hline &1&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrrr}3&1&-12&\color{orangered}{ 0 }&-69&192\\& & 3& \color{orangered}{-27} & & \\ \hline &1&-9&\color{orangered}{-27}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ -81 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&0&-69&192\\& & 3& -27& \color{blue}{-81} & \\ \hline &1&-9&\color{blue}{-27}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -69 } + \color{orangered}{ \left( -81 \right) } = \color{orangered}{ -150 } $
$$ \begin{array}{c|rrrrr}3&1&-12&0&\color{orangered}{ -69 }&192\\& & 3& -27& \color{orangered}{-81} & \\ \hline &1&-9&-27&\color{orangered}{-150}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ \left( -150 \right) } = \color{blue}{ -450 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{3}&1&-12&0&-69&192\\& & 3& -27& -81& \color{blue}{-450} \\ \hline &1&-9&-27&\color{blue}{-150}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 192 } + \color{orangered}{ \left( -450 \right) } = \color{orangered}{ -258 } $
$$ \begin{array}{c|rrrrr}3&1&-12&0&-69&\color{orangered}{ 192 }\\& & 3& -27& -81& \color{orangered}{-450} \\ \hline &\color{blue}{1}&\color{blue}{-9}&\color{blue}{-27}&\color{blue}{-150}&\color{orangered}{-258} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-9x^{2}-27x-150 } $ with a remainder of $ \color{red}{ -258 } $.