Divide using synthetic division $$ ( 2x^{3}-3x^{2}+8x+5 ) \div ( 2x+1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&2&-3&8&5\\& & -2& 5& \color{black}{-13} \\ \hline &\color{blue}{2}&\color{blue}{-5}&\color{blue}{13}&\color{orangered}{-8} \end{array} $$The solution is:
$$ \frac{ 2x^{3}-3x^{2}+8x+5 }{ x+1 } = \color{blue}{2x^{2}-5x+13} \color{red}{~-~} \frac{ \color{red}{ 8 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-3&8&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ 2 }&-3&8&5\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-3&8&5\\& & \color{blue}{-2} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-1&2&\color{orangered}{ -3 }&8&5\\& & \color{orangered}{-2} & & \\ \hline &2&\color{orangered}{-5}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-3&8&5\\& & -2& \color{blue}{5} & \\ \hline &2&\color{blue}{-5}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ 5 } = \color{orangered}{ 13 } $
$$ \begin{array}{c|rrrr}-1&2&-3&\color{orangered}{ 8 }&5\\& & -2& \color{orangered}{5} & \\ \hline &2&-5&\color{orangered}{13}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 13 } = \color{blue}{ -13 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&2&-3&8&5\\& & -2& 5& \color{blue}{-13} \\ \hline &2&-5&\color{blue}{13}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -13 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrr}-1&2&-3&8&\color{orangered}{ 5 }\\& & -2& 5& \color{orangered}{-13} \\ \hline &\color{blue}{2}&\color{blue}{-5}&\color{blue}{13}&\color{orangered}{-8} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-5x+13 } $ with a remainder of $ \color{red}{ -8 } $.