Divide using synthetic division $$ ( 3x^{6}-27x^{4}+x^{3}-10 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-3&3&0&-27&1&0&0&-10\\& & -9& 27& 0& -3& 9& \color{black}{-27} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{0}&\color{blue}{1}&\color{blue}{-3}&\color{blue}{9}&\color{orangered}{-37} \end{array} $$The solution is:
$$ \frac{ 3x^{6}-27x^{4}+x^{3}-10 }{ x+3 } = \color{blue}{3x^{5}-9x^{4}+x^{2}-3x+9} \color{red}{~-~} \frac{ \color{red}{ 37 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-3&\color{orangered}{ 3 }&0&-27&1&0&0&-10\\& & & & & & & \\ \hline &\color{orangered}{3}&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 3 } = \color{blue}{ -9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & \color{blue}{-9} & & & & & \\ \hline &\color{blue}{3}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -9 \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrrrr}-3&3&\color{orangered}{ 0 }&-27&1&0&0&-10\\& & \color{orangered}{-9} & & & & & \\ \hline &3&\color{orangered}{-9}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 27 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & -9& \color{blue}{27} & & & & \\ \hline &3&\color{blue}{-9}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 27 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrrrr}-3&3&0&\color{orangered}{ -27 }&1&0&0&-10\\& & -9& \color{orangered}{27} & & & & \\ \hline &3&-9&\color{orangered}{0}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & -9& 27& \color{blue}{0} & & & \\ \hline &3&-9&\color{blue}{0}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ 0 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrrr}-3&3&0&-27&\color{orangered}{ 1 }&0&0&-10\\& & -9& 27& \color{orangered}{0} & & & \\ \hline &3&-9&0&\color{orangered}{1}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & -9& 27& 0& \color{blue}{-3} & & \\ \hline &3&-9&0&\color{blue}{1}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrrrr}-3&3&0&-27&1&\color{orangered}{ 0 }&0&-10\\& & -9& 27& 0& \color{orangered}{-3} & & \\ \hline &3&-9&0&1&\color{orangered}{-3}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & -9& 27& 0& -3& \color{blue}{9} & \\ \hline &3&-9&0&1&\color{blue}{-3}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 9 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrrrr}-3&3&0&-27&1&0&\color{orangered}{ 0 }&-10\\& & -9& 27& 0& -3& \color{orangered}{9} & \\ \hline &3&-9&0&1&-3&\color{orangered}{9}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 9 } = \color{blue}{ -27 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-3}&3&0&-27&1&0&0&-10\\& & -9& 27& 0& -3& 9& \color{blue}{-27} \\ \hline &3&-9&0&1&-3&\color{blue}{9}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -27 \right) } = \color{orangered}{ -37 } $
$$ \begin{array}{c|rrrrrrr}-3&3&0&-27&1&0&0&\color{orangered}{ -10 }\\& & -9& 27& 0& -3& 9& \color{orangered}{-27} \\ \hline &\color{blue}{3}&\color{blue}{-9}&\color{blue}{0}&\color{blue}{1}&\color{blue}{-3}&\color{blue}{9}&\color{orangered}{-37} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 3x^{5}-9x^{4}+x^{2}-3x+9 } $ with a remainder of $ \color{red}{ -37 } $.