Divide using synthetic division $$ ( 12x^{3}-x^{2}-7x+2 ) \div ( x-4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}4&12&-1&-7&2\\& & 48& 188& \color{black}{724} \\ \hline &\color{blue}{12}&\color{blue}{47}&\color{blue}{181}&\color{orangered}{726} \end{array} $$The solution is:
$$ \dfrac{ 12x^{3}-x^{2}-7x+2 }{ x-4 } = \color{blue}{12x^{2}+47x+181} ~+~ \dfrac{ \color{red}{ 726 } }{ x-4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{4}&12&-1&-7&2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}4&\color{orangered}{ 12 }&-1&-7&2\\& & & & \\ \hline &\color{orangered}{12}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 12 } = \color{blue}{ 48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&12&-1&-7&2\\& & \color{blue}{48} & & \\ \hline &\color{blue}{12}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 48 } = \color{orangered}{ 47 } $
$$ \begin{array}{c|rrrr}4&12&\color{orangered}{ -1 }&-7&2\\& & \color{orangered}{48} & & \\ \hline &12&\color{orangered}{47}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 47 } = \color{blue}{ 188 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&12&-1&-7&2\\& & 48& \color{blue}{188} & \\ \hline &12&\color{blue}{47}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 188 } = \color{orangered}{ 181 } $
$$ \begin{array}{c|rrrr}4&12&-1&\color{orangered}{ -7 }&2\\& & 48& \color{orangered}{188} & \\ \hline &12&47&\color{orangered}{181}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 181 } = \color{blue}{ 724 } $.
$$ \begin{array}{c|rrrr}\color{blue}{4}&12&-1&-7&2\\& & 48& 188& \color{blue}{724} \\ \hline &12&47&\color{blue}{181}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 724 } = \color{orangered}{ 726 } $
$$ \begin{array}{c|rrrr}4&12&-1&-7&\color{orangered}{ 2 }\\& & 48& 188& \color{orangered}{724} \\ \hline &\color{blue}{12}&\color{blue}{47}&\color{blue}{181}&\color{orangered}{726} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 12x^{2}+47x+181 } $ with a remainder of $ \color{red}{ 726 } $.