Divide using synthetic division $$ ( 2x^{3}+8x+8 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&2&0&8&8\\& & -4& 8& \color{black}{-32} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{16}&\color{orangered}{-24} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+8x+8 }{ x+2 } = \color{blue}{2x^{2}-4x+16} \color{red}{~-~} \frac{ \color{red}{ 24 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&0&8&8\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 2 }&0&8&8\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&0&8&8\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-2&2&\color{orangered}{ 0 }&8&8\\& & \color{orangered}{-4} & & \\ \hline &2&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&0&8&8\\& & -4& \color{blue}{8} & \\ \hline &2&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 8 } = \color{orangered}{ 16 } $
$$ \begin{array}{c|rrrr}-2&2&0&\color{orangered}{ 8 }&8\\& & -4& \color{orangered}{8} & \\ \hline &2&-4&\color{orangered}{16}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 16 } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&0&8&8\\& & -4& 8& \color{blue}{-32} \\ \hline &2&-4&\color{blue}{16}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-2&2&0&8&\color{orangered}{ 8 }\\& & -4& 8& \color{orangered}{-32} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{16}&\color{orangered}{-24} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-4x+16 } $ with a remainder of $ \color{red}{ -24 } $.