Divide using synthetic division $$ ( -16x^{3}+23x^{2}-34x+24 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&-16&23&-34&24\\& & -32& -18& \color{black}{-104} \\ \hline &\color{blue}{-16}&\color{blue}{-9}&\color{blue}{-52}&\color{orangered}{-80} \end{array} $$The solution is:
$$ \frac{ -16x^{3}+23x^{2}-34x+24 }{ x-2 } = \color{blue}{-16x^{2}-9x-52} \color{red}{~-~} \frac{ \color{red}{ 80 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-16&23&-34&24\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ -16 }&23&-34&24\\& & & & \\ \hline &\color{orangered}{-16}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-16&23&-34&24\\& & \color{blue}{-32} & & \\ \hline &\color{blue}{-16}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}2&-16&\color{orangered}{ 23 }&-34&24\\& & \color{orangered}{-32} & & \\ \hline &-16&\color{orangered}{-9}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-16&23&-34&24\\& & -32& \color{blue}{-18} & \\ \hline &-16&\color{blue}{-9}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -52 } $
$$ \begin{array}{c|rrrr}2&-16&23&\color{orangered}{ -34 }&24\\& & -32& \color{orangered}{-18} & \\ \hline &-16&-9&\color{orangered}{-52}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -52 \right) } = \color{blue}{ -104 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&-16&23&-34&24\\& & -32& -18& \color{blue}{-104} \\ \hline &-16&-9&\color{blue}{-52}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -104 \right) } = \color{orangered}{ -80 } $
$$ \begin{array}{c|rrrr}2&-16&23&-34&\color{orangered}{ 24 }\\& & -32& -18& \color{orangered}{-104} \\ \hline &\color{blue}{-16}&\color{blue}{-9}&\color{blue}{-52}&\color{orangered}{-80} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -16x^{2}-9x-52 } $ with a remainder of $ \color{red}{ -80 } $.