Divide using synthetic division $$ ( 6x^{2}-8x+2 ) \div ( 7 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrr}0&6&-8&2\\& & 0& \color{black}{0} \\ \hline &\color{blue}{6}&\color{blue}{-8}&\color{orangered}{2} \end{array} $$The solution is:
$$ \dfrac{ 6x^{2}-8x+2 }{ x } = \color{blue}{6x-8} ~+~ \dfrac{ \color{red}{ 2 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrr}\color{blue}{0}&6&-8&2\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}0&\color{orangered}{ 6 }&-8&2\\& & & \\ \hline &\color{orangered}{6}&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 6 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&6&-8&2\\& & \color{blue}{0} & \\ \hline &\color{blue}{6}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 0 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrr}0&6&\color{orangered}{ -8 }&2\\& & \color{orangered}{0} & \\ \hline &6&\color{orangered}{-8}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrr}\color{blue}{0}&6&-8&2\\& & 0& \color{blue}{0} \\ \hline &6&\color{blue}{-8}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrr}0&6&-8&\color{orangered}{ 2 }\\& & 0& \color{orangered}{0} \\ \hline &\color{blue}{6}&\color{blue}{-8}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x-8 } $ with a remainder of $ \color{red}{ 2 } $.