The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&6&30&4&20\\& & -30& 0& \color{black}{-20} \\ \hline &\color{blue}{6}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{0} \end{array} $$The remainder when $ 6x^{3}+30x^{2}+4x+20 $ is divided by $ x+5 $ is $ \, \color{red}{ 0 } $.
We can find remainder using synthetic division method.
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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&30&4&20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 6 }&30&4&20\\& & & & \\ \hline &\color{orangered}{6}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 6 } = \color{blue}{ -30 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&30&4&20\\& & \color{blue}{-30} & & \\ \hline &\color{blue}{6}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 30 } + \color{orangered}{ \left( -30 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&6&\color{orangered}{ 30 }&4&20\\& & \color{orangered}{-30} & & \\ \hline &6&\color{orangered}{0}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 0 } = \color{blue}{ 0 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&30&4&20\\& & -30& \color{blue}{0} & \\ \hline &6&\color{blue}{0}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 0 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-5&6&30&\color{orangered}{ 4 }&20\\& & -30& \color{orangered}{0} & \\ \hline &6&0&\color{orangered}{4}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 4 } = \color{blue}{ -20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&6&30&4&20\\& & -30& 0& \color{blue}{-20} \\ \hline &6&0&\color{blue}{4}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 20 } + \color{orangered}{ \left( -20 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-5&6&30&4&\color{orangered}{ 20 }\\& & -30& 0& \color{orangered}{-20} \\ \hline &\color{blue}{6}&\color{blue}{0}&\color{blue}{4}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.