Divide using synthetic division $$ ( 2x^{4}-17x^{3}+30x^{2}+29x-20 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&2&-17&30&29&-20\\& & 8& -36& -24& \color{black}{20} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{-6}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-17x^{3}+30x^{2}+29x-20 }{ x-4 } = \color{blue}{2x^{3}-9x^{2}-6x+5} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&-17&30&29&-20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ 2 }&-17&30&29&-20\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 2 } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&-17&30&29&-20\\& & \color{blue}{8} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 8 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}4&2&\color{orangered}{ -17 }&30&29&-20\\& & \color{orangered}{8} & & & \\ \hline &2&\color{orangered}{-9}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&-17&30&29&-20\\& & 8& \color{blue}{-36} & & \\ \hline &2&\color{blue}{-9}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrr}4&2&-17&\color{orangered}{ 30 }&29&-20\\& & 8& \color{orangered}{-36} & & \\ \hline &2&-9&\color{orangered}{-6}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&-17&30&29&-20\\& & 8& -36& \color{blue}{-24} & \\ \hline &2&-9&\color{blue}{-6}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 29 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}4&2&-17&30&\color{orangered}{ 29 }&-20\\& & 8& -36& \color{orangered}{-24} & \\ \hline &2&-9&-6&\color{orangered}{5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ 5 } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&2&-17&30&29&-20\\& & 8& -36& -24& \color{blue}{20} \\ \hline &2&-9&-6&\color{blue}{5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 20 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&2&-17&30&29&\color{orangered}{ -20 }\\& & 8& -36& -24& \color{orangered}{20} \\ \hline &\color{blue}{2}&\color{blue}{-9}&\color{blue}{-6}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-9x^{2}-6x+5 } $ with a remainder of $ \color{red}{ 0 } $.