Divide using synthetic division $$ ( 4x^{7}-10x^{6}-5x^{3}+2x^{2}-4 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&0&-5&2&0&-4\\& & -8& 36& -72& 144& -278& 552& \color{black}{-1104} \\ \hline &\color{blue}{4}&\color{blue}{-18}&\color{blue}{36}&\color{blue}{-72}&\color{blue}{139}&\color{blue}{-276}&\color{blue}{552}&\color{orangered}{-1108} \end{array} $$The solution is:
$$ \frac{ 4x^{7}-10x^{6}-5x^{3}+2x^{2}-4 }{ x+2 } = \color{blue}{4x^{6}-18x^{5}+36x^{4}-72x^{3}+139x^{2}-276x+552} \color{red}{~-~} \frac{ \color{red}{ 1108 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}-2&\color{orangered}{ 4 }&-10&0&0&-5&2&0&-4\\& & & & & & & & \\ \hline &\color{orangered}{4}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & \color{blue}{-8} & & & & & & \\ \hline &\color{blue}{4}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -18 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&\color{orangered}{ -10 }&0&0&-5&2&0&-4\\& & \color{orangered}{-8} & & & & & & \\ \hline &4&\color{orangered}{-18}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -18 \right) } = \color{blue}{ 36 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& \color{blue}{36} & & & & & \\ \hline &4&\color{blue}{-18}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 36 } = \color{orangered}{ 36 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&\color{orangered}{ 0 }&0&-5&2&0&-4\\& & -8& \color{orangered}{36} & & & & & \\ \hline &4&-18&\color{orangered}{36}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 36 } = \color{blue}{ -72 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& 36& \color{blue}{-72} & & & & \\ \hline &4&-18&\color{blue}{36}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -72 \right) } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&\color{orangered}{ 0 }&-5&2&0&-4\\& & -8& 36& \color{orangered}{-72} & & & & \\ \hline &4&-18&36&\color{orangered}{-72}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -72 \right) } = \color{blue}{ 144 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& 36& -72& \color{blue}{144} & & & \\ \hline &4&-18&36&\color{blue}{-72}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 144 } = \color{orangered}{ 139 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&0&\color{orangered}{ -5 }&2&0&-4\\& & -8& 36& -72& \color{orangered}{144} & & & \\ \hline &4&-18&36&-72&\color{orangered}{139}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 139 } = \color{blue}{ -278 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& 36& -72& 144& \color{blue}{-278} & & \\ \hline &4&-18&36&-72&\color{blue}{139}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -278 \right) } = \color{orangered}{ -276 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&0&-5&\color{orangered}{ 2 }&0&-4\\& & -8& 36& -72& 144& \color{orangered}{-278} & & \\ \hline &4&-18&36&-72&139&\color{orangered}{-276}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -276 \right) } = \color{blue}{ 552 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& 36& -72& 144& -278& \color{blue}{552} & \\ \hline &4&-18&36&-72&139&\color{blue}{-276}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 552 } = \color{orangered}{ 552 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&0&-5&2&\color{orangered}{ 0 }&-4\\& & -8& 36& -72& 144& -278& \color{orangered}{552} & \\ \hline &4&-18&36&-72&139&-276&\color{orangered}{552}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 552 } = \color{blue}{ -1104 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{-2}&4&-10&0&0&-5&2&0&-4\\& & -8& 36& -72& 144& -278& 552& \color{blue}{-1104} \\ \hline &4&-18&36&-72&139&-276&\color{blue}{552}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -1104 \right) } = \color{orangered}{ -1108 } $
$$ \begin{array}{c|rrrrrrrr}-2&4&-10&0&0&-5&2&0&\color{orangered}{ -4 }\\& & -8& 36& -72& 144& -278& 552& \color{orangered}{-1104} \\ \hline &\color{blue}{4}&\color{blue}{-18}&\color{blue}{36}&\color{blue}{-72}&\color{blue}{139}&\color{blue}{-276}&\color{blue}{552}&\color{orangered}{-1108} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{6}-18x^{5}+36x^{4}-72x^{3}+139x^{2}-276x+552 } $ with a remainder of $ \color{red}{ -1108 } $.