Divide using synthetic division $$ ( 2x^{3}-2x^{2}+30x+134 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&2&-2&30&134\\& & -2& 4& \color{black}{-34} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{34}&\color{orangered}{100} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-2x^{2}+30x+134 }{ x+1 } = \color{blue}{2x^{2}-4x+34} ~+~ \frac{ \color{red}{ 100 } }{ x+1 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-2&30&134\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 2 }&-2&30&134\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 2 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-2&30&134\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrr}-1&2&\color{orangered}{ -2 }&30&134\\& & \color{orangered}{-2} & & \\ \hline &2&\color{orangered}{-4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-2&30&134\\& & -2& \color{blue}{4} & \\ \hline &2&\color{blue}{-4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ 4 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrr}-1&2&-2&\color{orangered}{ 30 }&134\\& & -2& \color{orangered}{4} & \\ \hline &2&-4&\color{orangered}{34}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 34 } = \color{blue}{ -34 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-2&30&134\\& & -2& 4& \color{blue}{-34} \\ \hline &2&-4&\color{blue}{34}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 134 } + \color{orangered}{ \left( -34 \right) } = \color{orangered}{ 100 } $
$$ \begin{array}{c|rrrr}-1&2&-2&30&\color{orangered}{ 134 }\\& & -2& 4& \color{orangered}{-34} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{34}&\color{orangered}{100} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-4x+34 } $ with a remainder of $ \color{red}{ 100 } $.