Divide using synthetic division $$ ( 6x^{4}-17x^{3}+24x^{2}-34x+24 ) \div ( 3x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&6&-17&24&-34&24\\& & 24& 28& 208& \color{black}{696} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{52}&\color{blue}{174}&\color{orangered}{720} \end{array} $$The solution is:
$$ \frac{ 6x^{4}-17x^{3}+24x^{2}-34x+24 }{ x-4 } = \color{blue}{6x^{3}+7x^{2}+52x+174} ~+~ \frac{ \color{red}{ 720 } }{ x-4 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&-17&24&-34&24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 6 }&-17&24&-34&24\\& & & & & \\ \hline &\color{orangered}{6}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 6 } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&-17&24&-34&24\\& & \color{blue}{24} & & & \\ \hline &\color{blue}{6}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 24 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrr}4&6&\color{orangered}{ -17 }&24&-34&24\\& & \color{orangered}{24} & & & \\ \hline &6&\color{orangered}{7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 7 } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&-17&24&-34&24\\& & 24& \color{blue}{28} & & \\ \hline &6&\color{blue}{7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 28 } = \color{orangered}{ 52 } $
$$ \begin{array}{c|rrrrr}4&6&-17&\color{orangered}{ 24 }&-34&24\\& & 24& \color{orangered}{28} & & \\ \hline &6&7&\color{orangered}{52}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 52 } = \color{blue}{ 208 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&-17&24&-34&24\\& & 24& 28& \color{blue}{208} & \\ \hline &6&7&\color{blue}{52}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 208 } = \color{orangered}{ 174 } $
$$ \begin{array}{c|rrrrr}4&6&-17&24&\color{orangered}{ -34 }&24\\& & 24& 28& \color{orangered}{208} & \\ \hline &6&7&52&\color{orangered}{174}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 174 } = \color{blue}{ 696 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&6&-17&24&-34&24\\& & 24& 28& 208& \color{blue}{696} \\ \hline &6&7&52&\color{blue}{174}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ 696 } = \color{orangered}{ 720 } $
$$ \begin{array}{c|rrrrr}4&6&-17&24&-34&\color{orangered}{ 24 }\\& & 24& 28& 208& \color{orangered}{696} \\ \hline &\color{blue}{6}&\color{blue}{7}&\color{blue}{52}&\color{blue}{174}&\color{orangered}{720} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{3}+7x^{2}+52x+174 } $ with a remainder of $ \color{red}{ 720 } $.