Divide using synthetic division $$ ( -x^{3}-32x^{2}+12x+144 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&-1&-32&12&144\\& & 1& 31& \color{black}{-43} \\ \hline &\color{blue}{-1}&\color{blue}{-31}&\color{blue}{43}&\color{orangered}{101} \end{array} $$The solution is:
$$ \dfrac{ -x^{3}-32x^{2}+12x+144 }{ x+1 } = \color{blue}{-x^{2}-31x+43} ~+~ \dfrac{ \color{red}{ 101 } }{ x+1 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&-1&-32&12&144\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ -1 }&-32&12&144\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 1 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&-1&-32&12&144\\& & \color{blue}{1} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -32 } + \color{orangered}{ 1 } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrr}-1&-1&\color{orangered}{ -32 }&12&144\\& & \color{orangered}{1} & & \\ \hline &-1&\color{orangered}{-31}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -31 \right) } = \color{blue}{ 31 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&-1&-32&12&144\\& & 1& \color{blue}{31} & \\ \hline &-1&\color{blue}{-31}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 12 } + \color{orangered}{ 31 } = \color{orangered}{ 43 } $
$$ \begin{array}{c|rrrr}-1&-1&-32&\color{orangered}{ 12 }&144\\& & 1& \color{orangered}{31} & \\ \hline &-1&-31&\color{orangered}{43}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ 43 } = \color{blue}{ -43 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&-1&-32&12&144\\& & 1& 31& \color{blue}{-43} \\ \hline &-1&-31&\color{blue}{43}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 144 } + \color{orangered}{ \left( -43 \right) } = \color{orangered}{ 101 } $
$$ \begin{array}{c|rrrr}-1&-1&-32&12&\color{orangered}{ 144 }\\& & 1& 31& \color{orangered}{-43} \\ \hline &\color{blue}{-1}&\color{blue}{-31}&\color{blue}{43}&\color{orangered}{101} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}-31x+43 } $ with a remainder of $ \color{red}{ 101 } $.