Divide using synthetic division $$ ( x^{4}+8x^{3}-74x-10 ) \div ( x+6 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-6&1&8&0&-74&-10\\& & -6& -12& 72& \color{black}{12} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-12}&\color{blue}{-2}&\color{orangered}{2} \end{array} $$The solution is:
$$ \frac{ x^{4}+8x^{3}-74x-10 }{ x+6 } = \color{blue}{x^{3}+2x^{2}-12x-2} ~+~ \frac{ \color{red}{ 2 } }{ x+6 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 6 = 0 $ ( $ x = \color{blue}{ -6 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&8&0&-74&-10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-6&\color{orangered}{ 1 }&8&0&-74&-10\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 1 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&8&0&-74&-10\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-6&1&\color{orangered}{ 8 }&0&-74&-10\\& & \color{orangered}{-6} & & & \\ \hline &1&\color{orangered}{2}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ 2 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&8&0&-74&-10\\& & -6& \color{blue}{-12} & & \\ \hline &1&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-6&1&8&\color{orangered}{ 0 }&-74&-10\\& & -6& \color{orangered}{-12} & & \\ \hline &1&2&\color{orangered}{-12}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 72 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&8&0&-74&-10\\& & -6& -12& \color{blue}{72} & \\ \hline &1&2&\color{blue}{-12}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -74 } + \color{orangered}{ 72 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-6&1&8&0&\color{orangered}{ -74 }&-10\\& & -6& -12& \color{orangered}{72} & \\ \hline &1&2&-12&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -6 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-6}&1&8&0&-74&-10\\& & -6& -12& 72& \color{blue}{12} \\ \hline &1&2&-12&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 12 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-6&1&8&0&-74&\color{orangered}{ -10 }\\& & -6& -12& 72& \color{orangered}{12} \\ \hline &\color{blue}{1}&\color{blue}{2}&\color{blue}{-12}&\color{blue}{-2}&\color{orangered}{2} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+2x^{2}-12x-2 } $ with a remainder of $ \color{red}{ 2 } $.