Divide using synthetic division $$ ( x^{5}+9x^{3}-12x^{2}+16x-18 ) \div ( -2x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-5&1&0&9&-12&16&-18\\& & -5& 25& -170& 910& \color{black}{-4630} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{34}&\color{blue}{-182}&\color{blue}{926}&\color{orangered}{-4648} \end{array} $$The solution is:
$$ \frac{ x^{5}+9x^{3}-12x^{2}+16x-18 }{ x+5 } = \color{blue}{x^{4}-5x^{3}+34x^{2}-182x+926} \color{red}{~-~} \frac{ \color{red}{ 4648 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-5&\color{orangered}{ 1 }&0&9&-12&16&-18\\& & & & & & \\ \hline &\color{orangered}{1}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & \color{blue}{-5} & & & & \\ \hline &\color{blue}{1}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrrr}-5&1&\color{orangered}{ 0 }&9&-12&16&-18\\& & \color{orangered}{-5} & & & & \\ \hline &1&\color{orangered}{-5}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & -5& \color{blue}{25} & & & \\ \hline &1&\color{blue}{-5}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ 25 } = \color{orangered}{ 34 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&\color{orangered}{ 9 }&-12&16&-18\\& & -5& \color{orangered}{25} & & & \\ \hline &1&-5&\color{orangered}{34}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 34 } = \color{blue}{ -170 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & -5& 25& \color{blue}{-170} & & \\ \hline &1&-5&\color{blue}{34}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ \left( -170 \right) } = \color{orangered}{ -182 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&9&\color{orangered}{ -12 }&16&-18\\& & -5& 25& \color{orangered}{-170} & & \\ \hline &1&-5&34&\color{orangered}{-182}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -182 \right) } = \color{blue}{ 910 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & -5& 25& -170& \color{blue}{910} & \\ \hline &1&-5&34&\color{blue}{-182}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ 910 } = \color{orangered}{ 926 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&9&-12&\color{orangered}{ 16 }&-18\\& & -5& 25& -170& \color{orangered}{910} & \\ \hline &1&-5&34&-182&\color{orangered}{926}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 926 } = \color{blue}{ -4630 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-5}&1&0&9&-12&16&-18\\& & -5& 25& -170& 910& \color{blue}{-4630} \\ \hline &1&-5&34&-182&\color{blue}{926}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ \left( -4630 \right) } = \color{orangered}{ -4648 } $
$$ \begin{array}{c|rrrrrr}-5&1&0&9&-12&16&\color{orangered}{ -18 }\\& & -5& 25& -170& 910& \color{orangered}{-4630} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{34}&\color{blue}{-182}&\color{blue}{926}&\color{orangered}{-4648} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{4}-5x^{3}+34x^{2}-182x+926 } $ with a remainder of $ \color{red}{ -4648 } $.