Divide using synthetic division $$ ( -2x^{7}-3x^{2}+2 ) \div ( -2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&0&0&-3&0&2\\& & 0& 0& 0& 0& 0& 0& \color{black}{0} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{blue}{0}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ -2x^{7}-3x^{2}+2 }{ x } = \color{blue}{-2x^{6}-3x} ~+~ \frac{ \color{red}{ 2 } }{ x } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}0&\color{orangered}{ -2 }&0&0&0&0&-3&0&2\\& & & & & & & & \\ \hline &\color{orangered}{-2}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & \color{blue}{0} & & & & & & \\ \hline &\color{blue}{-2}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&\color{orangered}{ 0 }&0&0&0&-3&0&2\\& & \color{orangered}{0} & & & & & & \\ \hline &-2&\color{orangered}{0}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& \color{blue}{0} & & & & & \\ \hline &-2&\color{blue}{0}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&\color{orangered}{ 0 }&0&0&-3&0&2\\& & 0& \color{orangered}{0} & & & & & \\ \hline &-2&0&\color{orangered}{0}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& 0& \color{blue}{0} & & & & \\ \hline &-2&0&\color{blue}{0}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&\color{orangered}{ 0 }&0&-3&0&2\\& & 0& 0& \color{orangered}{0} & & & & \\ \hline &-2&0&0&\color{orangered}{0}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& 0& 0& \color{blue}{0} & & & \\ \hline &-2&0&0&\color{blue}{0}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&0&\color{orangered}{ 0 }&-3&0&2\\& & 0& 0& 0& \color{orangered}{0} & & & \\ \hline &-2&0&0&0&\color{orangered}{0}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& 0& 0& 0& \color{blue}{0} & & \\ \hline &-2&0&0&0&\color{blue}{0}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ 0 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&0&0&\color{orangered}{ -3 }&0&2\\& & 0& 0& 0& 0& \color{orangered}{0} & & \\ \hline &-2&0&0&0&0&\color{orangered}{-3}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& 0& 0& 0& 0& \color{blue}{0} & \\ \hline &-2&0&0&0&0&\color{blue}{-3}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&0&0&-3&\color{orangered}{ 0 }&2\\& & 0& 0& 0& 0& 0& \color{orangered}{0} & \\ \hline &-2&0&0&0&0&-3&\color{orangered}{0}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{0}&-2&0&0&0&0&-3&0&2\\& & 0& 0& 0& 0& 0& 0& \color{blue}{0} \\ \hline &-2&0&0&0&0&-3&\color{blue}{0}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 0 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrrrrr}0&-2&0&0&0&0&-3&0&\color{orangered}{ 2 }\\& & 0& 0& 0& 0& 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-2}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{0}&\color{blue}{-3}&\color{blue}{0}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{6}-3x } $ with a remainder of $ \color{red}{ 2 } $.