Divide using synthetic division $$ ( x^{5}+4x^{4}+17x^{3}-14x^{2}-68x-40 ) \div ( x-3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}3&1&4&17&-14&-68&-40\\& & 3& 21& 114& 300& \color{black}{696} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{38}&\color{blue}{100}&\color{blue}{232}&\color{orangered}{656} \end{array} $$The solution is:
$$ \frac{ x^{5}+4x^{4}+17x^{3}-14x^{2}-68x-40 }{ x-3 } = \color{blue}{x^{4}+7x^{3}+38x^{2}+100x+232} ~+~ \frac{ \color{red}{ 656 } }{ x-3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -3 = 0 $ ( $ x = \color{blue}{ 3 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}3&\color{orangered}{ 1 }&4&17&-14&-68&-40\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 1 } = \color{blue}{ 3 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & \color{blue}{3} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 3 } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrrrr}3&1&\color{orangered}{ 4 }&17&-14&-68&-40\\& & \color{orangered}{3} & & & & \\ \hline &1&\color{orangered}{7}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 7 } = \color{blue}{ 21 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & 3& \color{blue}{21} & & & \\ \hline &1&\color{blue}{7}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 21 } = \color{orangered}{ 38 } $
$$ \begin{array}{c|rrrrrr}3&1&4&\color{orangered}{ 17 }&-14&-68&-40\\& & 3& \color{orangered}{21} & & & \\ \hline &1&7&\color{orangered}{38}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 38 } = \color{blue}{ 114 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & 3& 21& \color{blue}{114} & & \\ \hline &1&7&\color{blue}{38}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -14 } + \color{orangered}{ 114 } = \color{orangered}{ 100 } $
$$ \begin{array}{c|rrrrrr}3&1&4&17&\color{orangered}{ -14 }&-68&-40\\& & 3& 21& \color{orangered}{114} & & \\ \hline &1&7&38&\color{orangered}{100}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 100 } = \color{blue}{ 300 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & 3& 21& 114& \color{blue}{300} & \\ \hline &1&7&38&\color{blue}{100}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -68 } + \color{orangered}{ 300 } = \color{orangered}{ 232 } $
$$ \begin{array}{c|rrrrrr}3&1&4&17&-14&\color{orangered}{ -68 }&-40\\& & 3& 21& 114& \color{orangered}{300} & \\ \hline &1&7&38&100&\color{orangered}{232}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 3 } \cdot \color{blue}{ 232 } = \color{blue}{ 696 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{3}&1&4&17&-14&-68&-40\\& & 3& 21& 114& 300& \color{blue}{696} \\ \hline &1&7&38&100&\color{blue}{232}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -40 } + \color{orangered}{ 696 } = \color{orangered}{ 656 } $
$$ \begin{array}{c|rrrrrr}3&1&4&17&-14&-68&\color{orangered}{ -40 }\\& & 3& 21& 114& 300& \color{orangered}{696} \\ \hline &\color{blue}{1}&\color{blue}{7}&\color{blue}{38}&\color{blue}{100}&\color{blue}{232}&\color{orangered}{656} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}+7x^{3}+38x^{2}+100x+232 } $ with a remainder of $ \color{red}{ 656 } $.