Divide using synthetic division $$ ( 3x^{4}-4x^{3}-6x^{2}+17x-8 ) \div ( 3x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&3&-4&-6&17&-8\\& & 12& 32& 104& \color{black}{484} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{blue}{26}&\color{blue}{121}&\color{orangered}{476} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-4x^{3}-6x^{2}+17x-8 }{ x-4 } = \color{blue}{3x^{3}+8x^{2}+26x+121} ~+~ \frac{ \color{red}{ 476 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&3&-4&-6&17&-8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 3 }&-4&-6&17&-8\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 3 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&3&-4&-6&17&-8\\& & \color{blue}{12} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 12 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrr}4&3&\color{orangered}{ -4 }&-6&17&-8\\& & \color{orangered}{12} & & & \\ \hline &3&\color{orangered}{8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&3&-4&-6&17&-8\\& & 12& \color{blue}{32} & & \\ \hline &3&\color{blue}{8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ 32 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrrr}4&3&-4&\color{orangered}{ -6 }&17&-8\\& & 12& \color{orangered}{32} & & \\ \hline &3&8&\color{orangered}{26}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 26 } = \color{blue}{ 104 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&3&-4&-6&17&-8\\& & 12& 32& \color{blue}{104} & \\ \hline &3&8&\color{blue}{26}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 104 } = \color{orangered}{ 121 } $
$$ \begin{array}{c|rrrrr}4&3&-4&-6&\color{orangered}{ 17 }&-8\\& & 12& 32& \color{orangered}{104} & \\ \hline &3&8&26&\color{orangered}{121}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 121 } = \color{blue}{ 484 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&3&-4&-6&17&-8\\& & 12& 32& 104& \color{blue}{484} \\ \hline &3&8&26&\color{blue}{121}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 484 } = \color{orangered}{ 476 } $
$$ \begin{array}{c|rrrrr}4&3&-4&-6&17&\color{orangered}{ -8 }\\& & 12& 32& 104& \color{orangered}{484} \\ \hline &\color{blue}{3}&\color{blue}{8}&\color{blue}{26}&\color{blue}{121}&\color{orangered}{476} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+8x^{2}+26x+121 } $ with a remainder of $ \color{red}{ 476 } $.