Divide using synthetic division $$ ( 6x^{3}-28x^{2}+19x+4 ) \div ( 3x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&6&-28&19&4\\& & 12& -32& \color{black}{-26} \\ \hline &\color{blue}{6}&\color{blue}{-16}&\color{blue}{-13}&\color{orangered}{-22} \end{array} $$The solution is:
$$ \frac{ 6x^{3}-28x^{2}+19x+4 }{ x-2 } = \color{blue}{6x^{2}-16x-13} \color{red}{~-~} \frac{ \color{red}{ 22 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&-28&19&4\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 6 }&-28&19&4\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 6 } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&-28&19&4\\& & \color{blue}{12} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -28 } + \color{orangered}{ 12 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrr}2&6&\color{orangered}{ -28 }&19&4\\& & \color{orangered}{12} & & \\ \hline &6&\color{orangered}{-16}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&-28&19&4\\& & 12& \color{blue}{-32} & \\ \hline &6&\color{blue}{-16}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}2&6&-28&\color{orangered}{ 19 }&4\\& & 12& \color{orangered}{-32} & \\ \hline &6&-16&\color{orangered}{-13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ -26 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&6&-28&19&4\\& & 12& -32& \color{blue}{-26} \\ \hline &6&-16&\color{blue}{-13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -26 \right) } = \color{orangered}{ -22 } $
$$ \begin{array}{c|rrrr}2&6&-28&19&\color{orangered}{ 4 }\\& & 12& -32& \color{orangered}{-26} \\ \hline &\color{blue}{6}&\color{blue}{-16}&\color{blue}{-13}&\color{orangered}{-22} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{2}-16x-13 } $ with a remainder of $ \color{red}{ -22 } $.