The synthetic division table is:
$$ \begin{array}{c|rrrrr}0&3&-2&-7&4&2\\& & 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{-7}&\color{blue}{4}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ 3x^{4}-2x^{3}-7x^{2}+4x+2 }{ x } = \color{blue}{3x^{3}-2x^{2}-7x+4} ~+~ \frac{ \color{red}{ 2 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&-7&4&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}0&\color{orangered}{ 3 }&-2&-7&4&2\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 3 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&-7&4&2\\& & \color{blue}{0} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 0 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}0&3&\color{orangered}{ -2 }&-7&4&2\\& & \color{orangered}{0} & & & \\ \hline &3&\color{orangered}{-2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&-7&4&2\\& & 0& \color{blue}{0} & & \\ \hline &3&\color{blue}{-2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -7 } + \color{orangered}{ 0 } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}0&3&-2&\color{orangered}{ -7 }&4&2\\& & 0& \color{orangered}{0} & & \\ \hline &3&-2&\color{orangered}{-7}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&-7&4&2\\& & 0& 0& \color{blue}{0} & \\ \hline &3&-2&\color{blue}{-7}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}0&3&-2&-7&\color{orangered}{ 4 }&2\\& & 0& 0& \color{orangered}{0} & \\ \hline &3&-2&-7&\color{orangered}{4}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 4 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{0}&3&-2&-7&4&2\\& & 0& 0& 0& \color{blue}{0} \\ \hline &3&-2&-7&\color{blue}{4}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}0&3&-2&-7&4&\color{orangered}{ 2 }\\& & 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{3}&\color{blue}{-2}&\color{blue}{-7}&\color{blue}{4}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{3}-2x^{2}-7x+4 } $ with a remainder of $ \color{red}{ 2 } $.