Divide using synthetic division $$ ( 6x^{5}+x^{4}-35x^{3}-10x^{2}+44x+24 ) \div ( x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&6&1&-35&-10&44&24\\& & -18& 51& -48& 174& \color{black}{-654} \\ \hline &\color{blue}{6}&\color{blue}{-17}&\color{blue}{16}&\color{blue}{-58}&\color{blue}{218}&\color{orangered}{-630} \end{array} $$The solution is:
$$ \frac{ 6x^{5}+x^{4}-35x^{3}-10x^{2}+44x+24 }{ x+3 } = \color{blue}{6x^{4}-17x^{3}+16x^{2}-58x+218} \color{red}{~-~} \frac{ \color{red}{ 630 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 6 }&1&-35&-10&44&24\\& & & & & & \\ \hline &\color{orangered}{6}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 6 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & \color{blue}{-18} & & & & \\ \hline &\color{blue}{6}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrrr}-3&6&\color{orangered}{ 1 }&-35&-10&44&24\\& & \color{orangered}{-18} & & & & \\ \hline &6&\color{orangered}{-17}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 51 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & -18& \color{blue}{51} & & & \\ \hline &6&\color{blue}{-17}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -35 } + \color{orangered}{ 51 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrrrr}-3&6&1&\color{orangered}{ -35 }&-10&44&24\\& & -18& \color{orangered}{51} & & & \\ \hline &6&-17&\color{orangered}{16}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 16 } = \color{blue}{ -48 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & -18& 51& \color{blue}{-48} & & \\ \hline &6&-17&\color{blue}{16}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -48 \right) } = \color{orangered}{ -58 } $
$$ \begin{array}{c|rrrrrr}-3&6&1&-35&\color{orangered}{ -10 }&44&24\\& & -18& 51& \color{orangered}{-48} & & \\ \hline &6&-17&16&\color{orangered}{-58}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -58 \right) } = \color{blue}{ 174 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & -18& 51& -48& \color{blue}{174} & \\ \hline &6&-17&16&\color{blue}{-58}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 44 } + \color{orangered}{ 174 } = \color{orangered}{ 218 } $
$$ \begin{array}{c|rrrrrr}-3&6&1&-35&-10&\color{orangered}{ 44 }&24\\& & -18& 51& -48& \color{orangered}{174} & \\ \hline &6&-17&16&-58&\color{orangered}{218}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 218 } = \color{blue}{ -654 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&6&1&-35&-10&44&24\\& & -18& 51& -48& 174& \color{blue}{-654} \\ \hline &6&-17&16&-58&\color{blue}{218}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 24 } + \color{orangered}{ \left( -654 \right) } = \color{orangered}{ -630 } $
$$ \begin{array}{c|rrrrrr}-3&6&1&-35&-10&44&\color{orangered}{ 24 }\\& & -18& 51& -48& 174& \color{orangered}{-654} \\ \hline &\color{blue}{6}&\color{blue}{-17}&\color{blue}{16}&\color{blue}{-58}&\color{blue}{218}&\color{orangered}{-630} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 6x^{4}-17x^{3}+16x^{2}-58x+218 } $ with a remainder of $ \color{red}{ -630 } $.