Divide using synthetic division $$ ( 8x^{5}-4x^{4}+48x^{3}-24x^{2}-2 ) \div ( 8x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}4&8&-4&48&-24&0&-2\\& & 32& 112& 640& 2464& \color{black}{9856} \\ \hline &\color{blue}{8}&\color{blue}{28}&\color{blue}{160}&\color{blue}{616}&\color{blue}{2464}&\color{orangered}{9854} \end{array} $$The solution is:
$$ \frac{ 8x^{5}-4x^{4}+48x^{3}-24x^{2}-2 }{ x-4 } = \color{blue}{8x^{4}+28x^{3}+160x^{2}+616x+2464} ~+~ \frac{ \color{red}{ 9854 } }{ x-4 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -4 = 0 $ ( $ x = \color{blue}{ 4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}4&\color{orangered}{ 8 }&-4&48&-24&0&-2\\& & & & & & \\ \hline &\color{orangered}{8}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 8 } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & \color{blue}{32} & & & & \\ \hline &\color{blue}{8}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 32 } = \color{orangered}{ 28 } $
$$ \begin{array}{c|rrrrrr}4&8&\color{orangered}{ -4 }&48&-24&0&-2\\& & \color{orangered}{32} & & & & \\ \hline &8&\color{orangered}{28}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 28 } = \color{blue}{ 112 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & 32& \color{blue}{112} & & & \\ \hline &8&\color{blue}{28}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 48 } + \color{orangered}{ 112 } = \color{orangered}{ 160 } $
$$ \begin{array}{c|rrrrrr}4&8&-4&\color{orangered}{ 48 }&-24&0&-2\\& & 32& \color{orangered}{112} & & & \\ \hline &8&28&\color{orangered}{160}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 160 } = \color{blue}{ 640 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & 32& 112& \color{blue}{640} & & \\ \hline &8&28&\color{blue}{160}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 640 } = \color{orangered}{ 616 } $
$$ \begin{array}{c|rrrrrr}4&8&-4&48&\color{orangered}{ -24 }&0&-2\\& & 32& 112& \color{orangered}{640} & & \\ \hline &8&28&160&\color{orangered}{616}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 616 } = \color{blue}{ 2464 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & 32& 112& 640& \color{blue}{2464} & \\ \hline &8&28&160&\color{blue}{616}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 2464 } = \color{orangered}{ 2464 } $
$$ \begin{array}{c|rrrrrr}4&8&-4&48&-24&\color{orangered}{ 0 }&-2\\& & 32& 112& 640& \color{orangered}{2464} & \\ \hline &8&28&160&616&\color{orangered}{2464}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 2464 } = \color{blue}{ 9856 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{4}&8&-4&48&-24&0&-2\\& & 32& 112& 640& 2464& \color{blue}{9856} \\ \hline &8&28&160&616&\color{blue}{2464}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 9856 } = \color{orangered}{ 9854 } $
$$ \begin{array}{c|rrrrrr}4&8&-4&48&-24&0&\color{orangered}{ -2 }\\& & 32& 112& 640& 2464& \color{orangered}{9856} \\ \hline &\color{blue}{8}&\color{blue}{28}&\color{blue}{160}&\color{blue}{616}&\color{blue}{2464}&\color{orangered}{9854} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{4}+28x^{3}+160x^{2}+616x+2464 } $ with a remainder of $ \color{red}{ 9854 } $.