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