Divide using synthetic division $$ ( 5x^{3}-13x^{2}+9x+5 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-2&5&-13&9&5\\& & -10& 46& \color{black}{-110} \\ \hline &\color{blue}{5}&\color{blue}{-23}&\color{blue}{55}&\color{orangered}{-105} \end{array} $$The solution is:
$$ \frac{ 5x^{3}-13x^{2}+9x+5 }{ x+2 } = \color{blue}{5x^{2}-23x+55} \color{red}{~-~} \frac{ \color{red}{ 105 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-13&9&5\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-2&\color{orangered}{ 5 }&-13&9&5\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-13&9&5\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -23 } $
$$ \begin{array}{c|rrrr}-2&5&\color{orangered}{ -13 }&9&5\\& & \color{orangered}{-10} & & \\ \hline &5&\color{orangered}{-23}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -23 \right) } = \color{blue}{ 46 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-13&9&5\\& & -10& \color{blue}{46} & \\ \hline &5&\color{blue}{-23}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 46 } = \color{orangered}{ 55 } $
$$ \begin{array}{c|rrrr}-2&5&-13&\color{orangered}{ 9 }&5\\& & -10& \color{orangered}{46} & \\ \hline &5&-23&\color{orangered}{55}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 55 } = \color{blue}{ -110 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-2}&5&-13&9&5\\& & -10& 46& \color{blue}{-110} \\ \hline &5&-23&\color{blue}{55}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -110 \right) } = \color{orangered}{ -105 } $
$$ \begin{array}{c|rrrr}-2&5&-13&9&\color{orangered}{ 5 }\\& & -10& 46& \color{orangered}{-110} \\ \hline &\color{blue}{5}&\color{blue}{-23}&\color{blue}{55}&\color{orangered}{-105} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}-23x+55 } $ with a remainder of $ \color{red}{ -105 } $.