Divide using synthetic division $$ ( 2x^{4}+34x^{2}-11x+20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&2&0&34&-11&20\\& & -8& 32& -264& \color{black}{1100} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{66}&\color{blue}{-275}&\color{orangered}{1120} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+34x^{2}-11x+20 }{ x+4 } = \color{blue}{2x^{3}-8x^{2}+66x-275} ~+~ \frac{ \color{red}{ 1120 } }{ x+4 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&0&34&-11&20\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ 2 }&0&34&-11&20\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&0&34&-11&20\\& & \color{blue}{-8} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-4&2&\color{orangered}{ 0 }&34&-11&20\\& & \color{orangered}{-8} & & & \\ \hline &2&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&0&34&-11&20\\& & -8& \color{blue}{32} & & \\ \hline &2&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ 32 } = \color{orangered}{ 66 } $
$$ \begin{array}{c|rrrrr}-4&2&0&\color{orangered}{ 34 }&-11&20\\& & -8& \color{orangered}{32} & & \\ \hline &2&-8&\color{orangered}{66}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 66 } = \color{blue}{ -264 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&0&34&-11&20\\& & -8& 32& \color{blue}{-264} & \\ \hline &2&-8&\color{blue}{66}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ \left( -264 \right) } = \color{orangered}{ -275 } $
$$ \begin{array}{c|rrrrr}-4&2&0&34&\color{orangered}{ -11 }&20\\& & -8& 32& \color{orangered}{-264} & \\ \hline &2&-8&66&\color{orangered}{-275}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -275 \right) } = \color{blue}{ 1100 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&2&0&34&-11&20\\& & -8& 32& -264& \color{blue}{1100} \\ \hline &2&-8&66&\color{blue}{-275}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ 1100 } = \color{orangered}{ 1120 } $
$$ \begin{array}{c|rrrrr}-4&2&0&34&-11&\color{orangered}{ 20 }\\& & -8& 32& -264& \color{orangered}{1100} \\ \hline &\color{blue}{2}&\color{blue}{-8}&\color{blue}{66}&\color{blue}{-275}&\color{orangered}{1120} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-8x^{2}+66x-275 } $ with a remainder of $ \color{red}{ 1120 } $.