The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&3&-2&-18&-13&-18\\& & -6& 16& 4& \color{black}{18} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$The remainder when $ 3x^{4}-2x^{3}-18x^{2}-13x-18 $ is divided by $ x+2 $ 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 + 2 = 0 $ ( $ x = \color{blue}{ -2 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-2&-18&-13&-18\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 3 }&-2&-18&-13&-18\\& & & & & \\ \hline &\color{orangered}{3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 3 } = \color{blue}{ -6 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-2&-18&-13&-18\\& & \color{blue}{-6} & & & \\ \hline &\color{blue}{3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ \left( -6 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-2&3&\color{orangered}{ -2 }&-18&-13&-18\\& & \color{orangered}{-6} & & & \\ \hline &3&\color{orangered}{-8}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-2&-18&-13&-18\\& & -6& \color{blue}{16} & & \\ \hline &3&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 16 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-2&3&-2&\color{orangered}{ -18 }&-13&-18\\& & -6& \color{orangered}{16} & & \\ \hline &3&-8&\color{orangered}{-2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-2&-18&-13&-18\\& & -6& 16& \color{blue}{4} & \\ \hline &3&-8&\color{blue}{-2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 4 } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrrr}-2&3&-2&-18&\color{orangered}{ -13 }&-18\\& & -6& 16& \color{orangered}{4} & \\ \hline &3&-8&-2&\color{orangered}{-9}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&3&-2&-18&-13&-18\\& & -6& 16& 4& \color{blue}{18} \\ \hline &3&-8&-2&\color{blue}{-9}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -18 } + \color{orangered}{ 18 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&3&-2&-18&-13&\color{orangered}{ -18 }\\& & -6& 16& 4& \color{orangered}{18} \\ \hline &\color{blue}{3}&\color{blue}{-8}&\color{blue}{-2}&\color{blue}{-9}&\color{orangered}{0} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ 0 }\right) $.