The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&9&16&-36&-80\\& & -5& -20& 20& \color{black}{80} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{-16}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ x^{4}+9x^{3}+16x^{2}-36x-80 }{ x+5 } = \color{blue}{x^{3}+4x^{2}-4x-16} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&9&16&-36&-80\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&9&16&-36&-80\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&9&16&-36&-80\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 9 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ 9 }&16&-36&-80\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&9&16&-36&-80\\& & -5& \color{blue}{-20} & & \\ \hline &1&\color{blue}{4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}-5&1&9&\color{orangered}{ 16 }&-36&-80\\& & -5& \color{orangered}{-20} & & \\ \hline &1&4&\color{orangered}{-4}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&9&16&-36&-80\\& & -5& -20& \color{blue}{20} & \\ \hline &1&4&\color{blue}{-4}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -36 } + \color{orangered}{ 20 } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrr}-5&1&9&16&\color{orangered}{ -36 }&-80\\& & -5& -20& \color{orangered}{20} & \\ \hline &1&4&-4&\color{orangered}{-16}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 80 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&9&16&-36&-80\\& & -5& -20& 20& \color{blue}{80} \\ \hline &1&4&-4&\color{blue}{-16}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -80 } + \color{orangered}{ 80 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-5&1&9&16&-36&\color{orangered}{ -80 }\\& & -5& -20& 20& \color{orangered}{80} \\ \hline &\color{blue}{1}&\color{blue}{4}&\color{blue}{-4}&\color{blue}{-16}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}+4x^{2}-4x-16 } $ with a remainder of $ \color{red}{ 0 } $.