Divide using synthetic division $$ ( 5x^{3}+32x^{2}+11x-120 ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&5&32&11&-120\\& & -25& -35& \color{black}{120} \\ \hline &\color{blue}{5}&\color{blue}{7}&\color{blue}{-24}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ 5x^{3}+32x^{2}+11x-120 }{ x+5 } = \color{blue}{5x^{2}+7x-24} $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&32&11&-120\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 5 }&32&11&-120\\& & & & \\ \hline &\color{orangered}{5}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 5 } = \color{blue}{ -25 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&32&11&-120\\& & \color{blue}{-25} & & \\ \hline &\color{blue}{5}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 32 } + \color{orangered}{ \left( -25 \right) } = \color{orangered}{ 7 } $
$$ \begin{array}{c|rrrr}-5&5&\color{orangered}{ 32 }&11&-120\\& & \color{orangered}{-25} & & \\ \hline &5&\color{orangered}{7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 7 } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&32&11&-120\\& & -25& \color{blue}{-35} & \\ \hline &5&\color{blue}{7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ -24 } $
$$ \begin{array}{c|rrrr}-5&5&32&\color{orangered}{ 11 }&-120\\& & -25& \color{orangered}{-35} & \\ \hline &5&7&\color{orangered}{-24}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -24 \right) } = \color{blue}{ 120 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&5&32&11&-120\\& & -25& -35& \color{blue}{120} \\ \hline &5&7&\color{blue}{-24}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -120 } + \color{orangered}{ 120 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&5&32&11&\color{orangered}{ -120 }\\& & -25& -35& \color{orangered}{120} \\ \hline &\color{blue}{5}&\color{blue}{7}&\color{blue}{-24}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{2}+7x-24 } $ with a remainder of $ \color{red}{ 0 } $.