Divide using synthetic division $$ ( -24x^{3}+12x^{2}+34 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-24&12&0&34\\& & -48& -72& \color{black}{-144} \\ \hline &\color{blue}{-24}&\color{blue}{-36}&\color{blue}{-72}&\color{orangered}{-110} \end{array} $$The solution is:
$$ \frac{ -24x^{3}+12x^{2}+34 }{ x-2 } = \color{blue}{-24x^{2}-36x-72} \color{red}{~-~} \frac{ \color{red}{ 110 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-24&12&0&34\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -24 }&12&0&34\\& & & & \\ \hline &\color{orangered}{-24}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-24&12&0&34\\& & \color{blue}{-48} & & \\ \hline &\color{blue}{-24}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -36 } $
$$ \begin{array}{c|rrrr}2&-24&\color{orangered}{ 12 }&0&34\\& & \color{orangered}{-48} & & \\ \hline &-24&\color{orangered}{-36}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -36 \right) } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-24&12&0&34\\& & -48& \color{blue}{-72} & \\ \hline &-24&\color{blue}{-36}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrr}2&-24&12&\color{orangered}{ 0 }&34\\& & -48& \color{orangered}{-72} & \\ \hline &-24&-36&\color{orangered}{-72}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -72 \right) } = \color{blue}{ -144 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-24&12&0&34\\& & -48& -72& \color{blue}{-144} \\ \hline &-24&-36&\color{blue}{-72}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ \left( -144 \right) } = \color{orangered}{ -110 } $
$$ \begin{array}{c|rrrr}2&-24&12&0&\color{orangered}{ 34 }\\& & -48& -72& \color{orangered}{-144} \\ \hline &\color{blue}{-24}&\color{blue}{-36}&\color{blue}{-72}&\color{orangered}{-110} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -24x^{2}-36x-72 } $ with a remainder of $ \color{red}{ -110 } $.