Divide using synthetic division $$ ( 3x^{4}+10x^{3}+4x^{2}-5x-2 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}2&3&10&4&-5&-2\\& & 6& 32& 72& \color{black}{134} \\ \hline &\color{blue}{3}&\color{blue}{16}&\color{blue}{36}&\color{blue}{67}&\color{orangered}{132} \end{array} $$The solution is:
$$ \frac{ 3x^{4}+10x^{3}+4x^{2}-5x-2 }{ x-2 } = \color{blue}{3x^{3}+16x^{2}+36x+67} ~+~ \frac{ \color{red}{ 132 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&10&4&-5&-2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}2&\color{orangered}{ 3 }&10&4&-5&-2\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 3 } = \color{blue}{ 6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&10&4&-5&-2\\& & \color{blue}{6} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 6 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrr}2&3&\color{orangered}{ 10 }&4&-5&-2\\& & \color{orangered}{6} & & & \\ \hline &3&\color{orangered}{16}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 16 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&10&4&-5&-2\\& & 6& \color{blue}{32} & & \\ \hline &3&\color{blue}{16}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 32 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrrr}2&3&10&\color{orangered}{ 4 }&-5&-2\\& & 6& \color{orangered}{32} & & \\ \hline &3&16&\color{orangered}{36}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 36 } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&10&4&-5&-2\\& & 6& 32& \color{blue}{72} & \\ \hline &3&16&\color{blue}{36}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 72 } = \color{orangered}{ 67 } $
$$ \begin{array}{c|rrrrr}2&3&10&4&\color{orangered}{ -5 }&-2\\& & 6& 32& \color{orangered}{72} & \\ \hline &3&16&36&\color{orangered}{67}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 67 } = \color{blue}{ 134 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{2}&3&10&4&-5&-2\\& & 6& 32& 72& \color{blue}{134} \\ \hline &3&16&36&\color{blue}{67}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 134 } = \color{orangered}{ 132 } $
$$ \begin{array}{c|rrrrr}2&3&10&4&-5&\color{orangered}{ -2 }\\& & 6& 32& 72& \color{orangered}{134} \\ \hline &\color{blue}{3}&\color{blue}{16}&\color{blue}{36}&\color{blue}{67}&\color{orangered}{132} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}+16x^{2}+36x+67 } $ with a remainder of $ \color{red}{ 132 } $.