Divide using synthetic division $$ ( 5x^{3}+6x+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&5&0&6&8\\& & -10& 20& \color{black}{-52} \\ \hline &\color{blue}{5}&\color{blue}{-10}&\color{blue}{26}&\color{orangered}{-44} \end{array} $$The solution is:
$$ \frac{ 5x^{3}+6x+8 }{ x+2 } = \color{blue}{5x^{2}-10x+26} \color{red}{~-~} \frac{ \color{red}{ 44 } }{ x+2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&0&6&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 5 }&0&6&8\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&0&6&8\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -10 } $
$$ \begin{array}{c|rrrr}-2&5&\color{orangered}{ 0 }&6&8\\& & \color{orangered}{-10} & & \\ \hline &5&\color{orangered}{-10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -10 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&0&6&8\\& & -10& \color{blue}{20} & \\ \hline &5&\color{blue}{-10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ 20 } = \color{orangered}{ 26 } $
$$ \begin{array}{c|rrrr}-2&5&0&\color{orangered}{ 6 }&8\\& & -10& \color{orangered}{20} & \\ \hline &5&-10&\color{orangered}{26}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 26 } = \color{blue}{ -52 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&0&6&8\\& & -10& 20& \color{blue}{-52} \\ \hline &5&-10&\color{blue}{26}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -52 \right) } = \color{orangered}{ -44 } $
$$ \begin{array}{c|rrrr}-2&5&0&6&\color{orangered}{ 8 }\\& & -10& 20& \color{orangered}{-52} \\ \hline &\color{blue}{5}&\color{blue}{-10}&\color{blue}{26}&\color{orangered}{-44} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-10x+26 } $ with a remainder of $ \color{red}{ -44 } $.