The synthetic division table is:
$$ \begin{array}{c|rrrr}12&1&0&-34&-12\\& & 12& 144& \color{black}{1320} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{110}&\color{orangered}{1308} \end{array} $$The remainder when $ x^{3}-34x-12 $ is divided by $ x-12 $ is $ \, \color{red}{ 1308 } $.
We can find remainder using synthetic division method.
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -12 = 0 $ ( $ x = \color{blue}{ 12 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{12}&1&0&-34&-12\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}12&\color{orangered}{ 1 }&0&-34&-12\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 1 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{12}&1&0&-34&-12\\& & \color{blue}{12} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 12 } = \color{orangered}{ 12 } $
$$ \begin{array}{c|rrrr}12&1&\color{orangered}{ 0 }&-34&-12\\& & \color{orangered}{12} & & \\ \hline &1&\color{orangered}{12}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 12 } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{12}&1&0&-34&-12\\& & 12& \color{blue}{144} & \\ \hline &1&\color{blue}{12}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 144 } = \color{orangered}{ 110 } $
$$ \begin{array}{c|rrrr}12&1&0&\color{orangered}{ -34 }&-12\\& & 12& \color{orangered}{144} & \\ \hline &1&12&\color{orangered}{110}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 12 } \cdot \color{blue}{ 110 } = \color{blue}{ 1320 } $.
$$ \begin{array}{c|rrrr}\color{blue}{12}&1&0&-34&-12\\& & 12& 144& \color{blue}{1320} \\ \hline &1&12&\color{blue}{110}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 1320 } = \color{orangered}{ 1308 } $
$$ \begin{array}{c|rrrr}12&1&0&-34&\color{orangered}{ -12 }\\& & 12& 144& \color{orangered}{1320} \\ \hline &\color{blue}{1}&\color{blue}{12}&\color{blue}{110}&\color{orangered}{1308} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 1308 }\right) $.