Divide using synthetic division $$ ( 2x^{3}-4x^{2}-8x-9 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&2&-4&-8&-9\\& & -4& 16& \color{black}{-16} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{-25} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-4x^{2}-8x-9 }{ x+2 } = \color{blue}{2x^{2}-8x+8} \color{red}{~-~} \frac{ \color{red}{ 25 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&-4&-8&-9\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 2 }&-4&-8&-9\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&-4&-8&-9\\& & \color{blue}{-4} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-2&2&\color{orangered}{ -4 }&-8&-9\\& & \color{orangered}{-4} & & \\ \hline &2&\color{orangered}{-8}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&-4&-8&-9\\& & -4& \color{blue}{16} & \\ \hline &2&\color{blue}{-8}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 16 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrr}-2&2&-4&\color{orangered}{ -8 }&-9\\& & -4& \color{orangered}{16} & \\ \hline &2&-8&\color{orangered}{8}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&2&-4&-8&-9\\& & -4& 16& \color{blue}{-16} \\ \hline &2&-8&\color{blue}{8}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-2&2&-4&-8&\color{orangered}{ -9 }\\& & -4& 16& \color{orangered}{-16} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{-25} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-8x+8 } $ with a remainder of $ \color{red}{ -25 } $.