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