Divide using synthetic division $$ ( -5x^{3}-34x^{2}-43x+10 ) \div ( -2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}0&-5&-34&-43&10\\& & 0& 0& \color{black}{0} \\ \hline &\color{blue}{-5}&\color{blue}{-34}&\color{blue}{-43}&\color{orangered}{10} \end{array} $$The solution is:
$$ \dfrac{ -5x^{3}-34x^{2}-43x+10 }{ x } = \color{blue}{-5x^{2}-34x-43} ~+~ \dfrac{ \color{red}{ 10 } }{ x } $$Step 1 : Write down the coefficients of the dividend into division table.Put the zero at the left.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&-34&-43&10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}0&\color{orangered}{ -5 }&-34&-43&10\\& & & & \\ \hline &\color{orangered}{-5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&-34&-43&10\\& & \color{blue}{0} & & \\ \hline &\color{blue}{-5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -34 } + \color{orangered}{ 0 } = \color{orangered}{ -34 } $
$$ \begin{array}{c|rrrr}0&-5&\color{orangered}{ -34 }&-43&10\\& & \color{orangered}{0} & & \\ \hline &-5&\color{orangered}{-34}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -34 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&-34&-43&10\\& & 0& \color{blue}{0} & \\ \hline &-5&\color{blue}{-34}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -43 } + \color{orangered}{ 0 } = \color{orangered}{ -43 } $
$$ \begin{array}{c|rrrr}0&-5&-34&\color{orangered}{ -43 }&10\\& & 0& \color{orangered}{0} & \\ \hline &-5&-34&\color{orangered}{-43}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 0 } \cdot \color{blue}{ \left( -43 \right) } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{0}&-5&-34&-43&10\\& & 0& 0& \color{blue}{0} \\ \hline &-5&-34&\color{blue}{-43}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ 0 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}0&-5&-34&-43&\color{orangered}{ 10 }\\& & 0& 0& \color{orangered}{0} \\ \hline &\color{blue}{-5}&\color{blue}{-34}&\color{blue}{-43}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -5x^{2}-34x-43 } $ with a remainder of $ \color{red}{ 10 } $.