Divide using synthetic division $$ ( 9x^{5}+37x^{4}-3x^{3}-27x^{2}+2 ) \div ( x+4 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-4&9&37&-3&-27&0&2\\& & -36& -4& 28& -4& \color{black}{16} \\ \hline &\color{blue}{9}&\color{blue}{1}&\color{blue}{-7}&\color{blue}{1}&\color{blue}{-4}&\color{orangered}{18} \end{array} $$The solution is:
$$ \dfrac{ 9x^{5}+37x^{4}-3x^{3}-27x^{2}+2 }{ x+4 } = \color{blue}{9x^{4}+x^{3}-7x^{2}+x-4} ~+~ \dfrac{ \color{red}{ 18 } }{ x+4 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-4&\color{orangered}{ 9 }&37&-3&-27&0&2\\& & & & & & \\ \hline &\color{orangered}{9}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 9 } = \color{blue}{ -36 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & \color{blue}{-36} & & & & \\ \hline &\color{blue}{9}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 37 } + \color{orangered}{ \left( -36 \right) } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-4&9&\color{orangered}{ 37 }&-3&-27&0&2\\& & \color{orangered}{-36} & & & & \\ \hline &9&\color{orangered}{1}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & -36& \color{blue}{-4} & & & \\ \hline &9&\color{blue}{1}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrrr}-4&9&37&\color{orangered}{ -3 }&-27&0&2\\& & -36& \color{orangered}{-4} & & & \\ \hline &9&1&\color{orangered}{-7}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 28 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & -36& -4& \color{blue}{28} & & \\ \hline &9&1&\color{blue}{-7}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -27 } + \color{orangered}{ 28 } = \color{orangered}{ 1 } $
$$ \begin{array}{c|rrrrrr}-4&9&37&-3&\color{orangered}{ -27 }&0&2\\& & -36& -4& \color{orangered}{28} & & \\ \hline &9&1&-7&\color{orangered}{1}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 1 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & -36& -4& 28& \color{blue}{-4} & \\ \hline &9&1&-7&\color{blue}{1}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrr}-4&9&37&-3&-27&\color{orangered}{ 0 }&2\\& & -36& -4& 28& \color{orangered}{-4} & \\ \hline &9&1&-7&1&\color{orangered}{-4}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-4}&9&37&-3&-27&0&2\\& & -36& -4& 28& -4& \color{blue}{16} \\ \hline &9&1&-7&1&\color{blue}{-4}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 16 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}-4&9&37&-3&-27&0&\color{orangered}{ 2 }\\& & -36& -4& 28& -4& \color{orangered}{16} \\ \hline &\color{blue}{9}&\color{blue}{1}&\color{blue}{-7}&\color{blue}{1}&\color{blue}{-4}&\color{orangered}{18} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{4}+x^{3}-7x^{2}+x-4 } $ with a remainder of $ \color{red}{ 18 } $.