Divide using synthetic division $$ ( 10x^{5}+3x^{4}+5x^{2}+6x-4 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&10&3&0&5&6&-4\\& & -30& 81& -243& 714& \color{black}{-2160} \\ \hline &\color{blue}{10}&\color{blue}{-27}&\color{blue}{81}&\color{blue}{-238}&\color{blue}{720}&\color{orangered}{-2164} \end{array} $$The solution is:
$$ \frac{ 10x^{5}+3x^{4}+5x^{2}+6x-4 }{ x+3 } = \color{blue}{10x^{4}-27x^{3}+81x^{2}-238x+720} \color{red}{~-~} \frac{ \color{red}{ 2164 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 10 }&3&0&5&6&-4\\& & & & & & \\ \hline &\color{orangered}{10}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 10 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & \color{blue}{-30} & & & & \\ \hline &\color{blue}{10}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ -27 } $
$$ \begin{array}{c|rrrrrr}-3&10&\color{orangered}{ 3 }&0&5&6&-4\\& & \color{orangered}{-30} & & & & \\ \hline &10&\color{orangered}{-27}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -27 \right) } = \color{blue}{ 81 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & -30& \color{blue}{81} & & & \\ \hline &10&\color{blue}{-27}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 81 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrrr}-3&10&3&\color{orangered}{ 0 }&5&6&-4\\& & -30& \color{orangered}{81} & & & \\ \hline &10&-27&\color{orangered}{81}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 81 } = \color{blue}{ -243 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & -30& 81& \color{blue}{-243} & & \\ \hline &10&-27&\color{blue}{81}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -243 \right) } = \color{orangered}{ -238 } $
$$ \begin{array}{c|rrrrrr}-3&10&3&0&\color{orangered}{ 5 }&6&-4\\& & -30& 81& \color{orangered}{-243} & & \\ \hline &10&-27&81&\color{orangered}{-238}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -238 \right) } = \color{blue}{ 714 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & -30& 81& -243& \color{blue}{714} & \\ \hline &10&-27&81&\color{blue}{-238}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 714 } = \color{orangered}{ 720 } $
$$ \begin{array}{c|rrrrrr}-3&10&3&0&5&\color{orangered}{ 6 }&-4\\& & -30& 81& -243& \color{orangered}{714} & \\ \hline &10&-27&81&-238&\color{orangered}{720}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 720 } = \color{blue}{ -2160 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&10&3&0&5&6&-4\\& & -30& 81& -243& 714& \color{blue}{-2160} \\ \hline &10&-27&81&-238&\color{blue}{720}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -2160 \right) } = \color{orangered}{ -2164 } $
$$ \begin{array}{c|rrrrrr}-3&10&3&0&5&6&\color{orangered}{ -4 }\\& & -30& 81& -243& 714& \color{orangered}{-2160} \\ \hline &\color{blue}{10}&\color{blue}{-27}&\color{blue}{81}&\color{blue}{-238}&\color{blue}{720}&\color{orangered}{-2164} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{4}-27x^{3}+81x^{2}-238x+720 } $ with a remainder of $ \color{red}{ -2164 } $.