Divide using synthetic division $$ ( 2x^{3}-15x^{2}+34x-21 ) \div ( x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&2&-15&34&-21\\& & -2& 17& \color{black}{-51} \\ \hline &\color{blue}{2}&\color{blue}{-17}&\color{blue}{51}&\color{orangered}{-72} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-15x^{2}+34x-21 }{ x+1 } = \color{blue}{2x^{2}-17x+51} \color{red}{~-~} \frac{ \color{red}{ 72 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-15&34&-21\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 2 }&-15&34&-21\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-15&34&-21\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -15 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrr}-1&2&\color{orangered}{ -15 }&34&-21\\& & \color{orangered}{-2} & & \\ \hline &2&\color{orangered}{-17}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 17 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-15&34&-21\\& & -2& \color{blue}{17} & \\ \hline &2&\color{blue}{-17}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 34 } + \color{orangered}{ 17 } = \color{orangered}{ 51 } $
$$ \begin{array}{c|rrrr}-1&2&-15&\color{orangered}{ 34 }&-21\\& & -2& \color{orangered}{17} & \\ \hline &2&-17&\color{orangered}{51}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 51 } = \color{blue}{ -51 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-15&34&-21\\& & -2& 17& \color{blue}{-51} \\ \hline &2&-17&\color{blue}{51}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -21 } + \color{orangered}{ \left( -51 \right) } = \color{orangered}{ -72 } $
$$ \begin{array}{c|rrrr}-1&2&-15&34&\color{orangered}{ -21 }\\& & -2& 17& \color{orangered}{-51} \\ \hline &\color{blue}{2}&\color{blue}{-17}&\color{blue}{51}&\color{orangered}{-72} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-17x+51 } $ with a remainder of $ \color{red}{ -72 } $.