Divide using synthetic division $$ ( -8x^{7}+5x^{3}-x-4 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&0&5&0&-1&-4\\& & -16& -32& -64& -128& -246& -492& \color{black}{-986} \\ \hline &\color{blue}{-8}&\color{blue}{-16}&\color{blue}{-32}&\color{blue}{-64}&\color{blue}{-123}&\color{blue}{-246}&\color{blue}{-493}&\color{orangered}{-990} \end{array} $$The solution is:
$$ \frac{ -8x^{7}+5x^{3}-x-4 }{ x-2 } = \color{blue}{-8x^{6}-16x^{5}-32x^{4}-64x^{3}-123x^{2}-246x-493} \color{red}{~-~} \frac{ \color{red}{ 990 } }{ x-2 } $$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.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & & & & & & & \\ \hline &&&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrrr}2&\color{orangered}{ -8 }&0&0&0&5&0&-1&-4\\& & & & & & & & \\ \hline &\color{orangered}{-8}&&&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & \color{blue}{-16} & & & & & & \\ \hline &\color{blue}{-8}&&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&\color{orangered}{ 0 }&0&0&5&0&-1&-4\\& & \color{orangered}{-16} & & & & & & \\ \hline &-8&\color{orangered}{-16}&&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ -32 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& \color{blue}{-32} & & & & & \\ \hline &-8&\color{blue}{-16}&&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -32 \right) } = \color{orangered}{ -32 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&\color{orangered}{ 0 }&0&5&0&-1&-4\\& & -16& \color{orangered}{-32} & & & & & \\ \hline &-8&-16&\color{orangered}{-32}&&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -32 \right) } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& -32& \color{blue}{-64} & & & & \\ \hline &-8&-16&\color{blue}{-32}&&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -64 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&\color{orangered}{ 0 }&5&0&-1&-4\\& & -16& -32& \color{orangered}{-64} & & & & \\ \hline &-8&-16&-32&\color{orangered}{-64}&&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -64 \right) } = \color{blue}{ -128 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& -32& -64& \color{blue}{-128} & & & \\ \hline &-8&-16&-32&\color{blue}{-64}&&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -128 \right) } = \color{orangered}{ -123 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&0&\color{orangered}{ 5 }&0&-1&-4\\& & -16& -32& -64& \color{orangered}{-128} & & & \\ \hline &-8&-16&-32&-64&\color{orangered}{-123}&&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -123 \right) } = \color{blue}{ -246 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& -32& -64& -128& \color{blue}{-246} & & \\ \hline &-8&-16&-32&-64&\color{blue}{-123}&&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -246 \right) } = \color{orangered}{ -246 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&0&5&\color{orangered}{ 0 }&-1&-4\\& & -16& -32& -64& -128& \color{orangered}{-246} & & \\ \hline &-8&-16&-32&-64&-123&\color{orangered}{-246}&& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -246 \right) } = \color{blue}{ -492 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& -32& -64& -128& -246& \color{blue}{-492} & \\ \hline &-8&-16&-32&-64&-123&\color{blue}{-246}&& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ \left( -492 \right) } = \color{orangered}{ -493 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&0&5&0&\color{orangered}{ -1 }&-4\\& & -16& -32& -64& -128& -246& \color{orangered}{-492} & \\ \hline &-8&-16&-32&-64&-123&-246&\color{orangered}{-493}& \end{array} $$Step 14 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ \left( -493 \right) } = \color{blue}{ -986 } $.
$$ \begin{array}{c|rrrrrrrr}\color{blue}{2}&-8&0&0&0&5&0&-1&-4\\& & -16& -32& -64& -128& -246& -492& \color{blue}{-986} \\ \hline &-8&-16&-32&-64&-123&-246&\color{blue}{-493}& \end{array} $$Step 15 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ \left( -986 \right) } = \color{orangered}{ -990 } $
$$ \begin{array}{c|rrrrrrrr}2&-8&0&0&0&5&0&-1&\color{orangered}{ -4 }\\& & -16& -32& -64& -128& -246& -492& \color{orangered}{-986} \\ \hline &\color{blue}{-8}&\color{blue}{-16}&\color{blue}{-32}&\color{blue}{-64}&\color{blue}{-123}&\color{blue}{-246}&\color{blue}{-493}&\color{orangered}{-990} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -8x^{6}-16x^{5}-32x^{4}-64x^{3}-123x^{2}-246x-493 } $ with a remainder of $ \color{red}{ -990 } $.