Divide using synthetic division $$ ( x^{5}-8x^{4}+22x^{3}-18x^{2}-19x+30 ) \div ( x ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}0&1&-8&22&-18&-19&30\\& & 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{22}&\color{blue}{-18}&\color{blue}{-19}&\color{orangered}{30} \end{array} $$The solution is:
$$ \dfrac{ x^{5}-8x^{4}+22x^{3}-18x^{2}-19x+30 }{ x } = \color{blue}{x^{4}-8x^{3}+22x^{2}-18x-19} ~+~ \dfrac{ \color{red}{ 30 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}0&\color{orangered}{ 1 }&-8&22&-18&-19&30\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 1 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & \color{blue}{0} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 0 } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}0&1&\color{orangered}{ -8 }&22&-18&-19&30\\& & \color{orangered}{0} & & & & \\ \hline &1&\color{orangered}{-8}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & 0& \color{blue}{0} & & & \\ \hline &1&\color{blue}{-8}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 22 } + \color{orangered}{ 0 } = \color{orangered}{ 22 } $
$$ \begin{array}{c|rrrrrr}0&1&-8&\color{orangered}{ 22 }&-18&-19&30\\& & 0& \color{orangered}{0} & & & \\ \hline &1&-8&\color{orangered}{22}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 22 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & 0& 0& \color{blue}{0} & & \\ \hline &1&-8&\color{blue}{22}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 0 } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrr}0&1&-8&22&\color{orangered}{ -18 }&-19&30\\& & 0& 0& \color{orangered}{0} & & \\ \hline &1&-8&22&\color{orangered}{-18}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & 0& 0& 0& \color{blue}{0} & \\ \hline &1&-8&22&\color{blue}{-18}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -19 } + \color{orangered}{ 0 } = \color{orangered}{ -19 } $
$$ \begin{array}{c|rrrrrr}0&1&-8&22&-18&\color{orangered}{ -19 }&30\\& & 0& 0& 0& \color{orangered}{0} & \\ \hline &1&-8&22&-18&\color{orangered}{-19}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -19 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{0}&1&-8&22&-18&-19&30\\& & 0& 0& 0& 0& \color{blue}{0} \\ \hline &1&-8&22&-18&\color{blue}{-19}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 0 } = \color{orangered}{ 30 } $
$$ \begin{array}{c|rrrrrr}0&1&-8&22&-18&-19&\color{orangered}{ 30 }\\& & 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{22}&\color{blue}{-18}&\color{blue}{-19}&\color{orangered}{30} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-8x^{3}+22x^{2}-18x-19 } $ with a remainder of $ \color{red}{ 30 } $.