Divide using synthetic division $$ ( 8x^{4}+22x^{3}-41x^{2}-x+12 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&8&22&-41&-1&12\\& & 32& 216& 700& \color{black}{2796} \\ \hline &\color{blue}{8}&\color{blue}{54}&\color{blue}{175}&\color{blue}{699}&\color{orangered}{2808} \end{array} $$The solution is:
$$ \frac{ 8x^{4}+22x^{3}-41x^{2}-x+12 }{ x-4 } = \color{blue}{8x^{3}+54x^{2}+175x+699} ~+~ \frac{ \color{red}{ 2808 } }{ x-4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&22&-41&-1&12\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 8 }&22&-41&-1&12\\& & & & & \\ \hline &\color{orangered}{8}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&22&-41&-1&12\\& & \color{blue}{32} & & & \\ \hline &\color{blue}{8}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 22 } + \color{orangered}{ 32 } = \color{orangered}{ 54 } $
$$ \begin{array}{c|rrrrr}4&8&\color{orangered}{ 22 }&-41&-1&12\\& & \color{orangered}{32} & & & \\ \hline &8&\color{orangered}{54}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 54 } = \color{blue}{ 216 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&22&-41&-1&12\\& & 32& \color{blue}{216} & & \\ \hline &8&\color{blue}{54}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -41 } + \color{orangered}{ 216 } = \color{orangered}{ 175 } $
$$ \begin{array}{c|rrrrr}4&8&22&\color{orangered}{ -41 }&-1&12\\& & 32& \color{orangered}{216} & & \\ \hline &8&54&\color{orangered}{175}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 175 } = \color{blue}{ 700 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&22&-41&-1&12\\& & 32& 216& \color{blue}{700} & \\ \hline &8&54&\color{blue}{175}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 700 } = \color{orangered}{ 699 } $
$$ \begin{array}{c|rrrrr}4&8&22&-41&\color{orangered}{ -1 }&12\\& & 32& 216& \color{orangered}{700} & \\ \hline &8&54&175&\color{orangered}{699}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 699 } = \color{blue}{ 2796 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&8&22&-41&-1&12\\& & 32& 216& 700& \color{blue}{2796} \\ \hline &8&54&175&\color{blue}{699}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 2796 } = \color{orangered}{ 2808 } $
$$ \begin{array}{c|rrrrr}4&8&22&-41&-1&\color{orangered}{ 12 }\\& & 32& 216& 700& \color{orangered}{2796} \\ \hline &\color{blue}{8}&\color{blue}{54}&\color{blue}{175}&\color{blue}{699}&\color{orangered}{2808} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 8x^{3}+54x^{2}+175x+699 } $ with a remainder of $ \color{red}{ 2808 } $.