The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&1&-14&71&42&-222\\& & -7& 147& -1526& \color{black}{10388} \\ \hline &\color{blue}{1}&\color{blue}{-21}&\color{blue}{218}&\color{blue}{-1484}&\color{orangered}{10166} \end{array} $$The solution is:
$$ \frac{ x^{4}-14x^{3}+71x^{2}+42x-222 }{ x+7 } = \color{blue}{x^{3}-21x^{2}+218x-1484} ~+~ \frac{ \color{red}{ 10166 } }{ x+7 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-14&71&42&-222\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 1 }&-14&71&42&-222\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 1 } = \color{blue}{ -7 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-14&71&42&-222\\& & \color{blue}{-7} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ \left( -7 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrrrr}-7&1&\color{orangered}{ -14 }&71&42&-222\\& & \color{orangered}{-7} & & & \\ \hline &1&\color{orangered}{-21}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -21 \right) } = \color{blue}{ 147 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-14&71&42&-222\\& & -7& \color{blue}{147} & & \\ \hline &1&\color{blue}{-21}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 71 } + \color{orangered}{ 147 } = \color{orangered}{ 218 } $
$$ \begin{array}{c|rrrrr}-7&1&-14&\color{orangered}{ 71 }&42&-222\\& & -7& \color{orangered}{147} & & \\ \hline &1&-21&\color{orangered}{218}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 218 } = \color{blue}{ -1526 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-14&71&42&-222\\& & -7& 147& \color{blue}{-1526} & \\ \hline &1&-21&\color{blue}{218}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 42 } + \color{orangered}{ \left( -1526 \right) } = \color{orangered}{ -1484 } $
$$ \begin{array}{c|rrrrr}-7&1&-14&71&\color{orangered}{ 42 }&-222\\& & -7& 147& \color{orangered}{-1526} & \\ \hline &1&-21&218&\color{orangered}{-1484}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -1484 \right) } = \color{blue}{ 10388 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&1&-14&71&42&-222\\& & -7& 147& -1526& \color{blue}{10388} \\ \hline &1&-21&218&\color{blue}{-1484}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -222 } + \color{orangered}{ 10388 } = \color{orangered}{ 10166 } $
$$ \begin{array}{c|rrrrr}-7&1&-14&71&42&\color{orangered}{ -222 }\\& & -7& 147& -1526& \color{orangered}{10388} \\ \hline &\color{blue}{1}&\color{blue}{-21}&\color{blue}{218}&\color{blue}{-1484}&\color{orangered}{10166} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-21x^{2}+218x-1484 } $ with a remainder of $ \color{red}{ 10166 } $.