The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-2&2&15&26&0&-8&14\\& & -4& -22& -8& 16& \color{black}{-16} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{4}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{-2} \end{array} $$The remainder when $ 2x^{5}+15x^{4}+26x^{3}-8x+14 $ is divided by $ x+2 $ is $ \, \color{red}{ -2 } $.
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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-2&\color{orangered}{ 2 }&15&26&0&-8&14\\& & & & & & \\ \hline &\color{orangered}{2}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 2 } = \color{blue}{ -4 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & \color{blue}{-4} & & & & \\ \hline &\color{blue}{2}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 15 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ 11 } $
$$ \begin{array}{c|rrrrrr}-2&2&\color{orangered}{ 15 }&26&0&-8&14\\& & \color{orangered}{-4} & & & & \\ \hline &2&\color{orangered}{11}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 11 } = \color{blue}{ -22 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & -4& \color{blue}{-22} & & & \\ \hline &2&\color{blue}{11}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -22 \right) } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrrrr}-2&2&15&\color{orangered}{ 26 }&0&-8&14\\& & -4& \color{orangered}{-22} & & & \\ \hline &2&11&\color{orangered}{4}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 4 } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & -4& -22& \color{blue}{-8} & & \\ \hline &2&11&\color{blue}{4}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrrr}-2&2&15&26&\color{orangered}{ 0 }&-8&14\\& & -4& -22& \color{orangered}{-8} & & \\ \hline &2&11&4&\color{orangered}{-8}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -8 \right) } = \color{blue}{ 16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & -4& -22& -8& \color{blue}{16} & \\ \hline &2&11&4&\color{blue}{-8}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ 16 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrr}-2&2&15&26&0&\color{orangered}{ -8 }&14\\& & -4& -22& -8& \color{orangered}{16} & \\ \hline &2&11&4&-8&\color{orangered}{8}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-2}&2&15&26&0&-8&14\\& & -4& -22& -8& 16& \color{blue}{-16} \\ \hline &2&11&4&-8&\color{blue}{8}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 14 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrrr}-2&2&15&26&0&-8&\color{orangered}{ 14 }\\& & -4& -22& -8& 16& \color{orangered}{-16} \\ \hline &\color{blue}{2}&\color{blue}{11}&\color{blue}{4}&\color{blue}{-8}&\color{blue}{8}&\color{orangered}{-2} \end{array} $$Remainder is the last entry in the bottom row $ \left(\color{red}{ -2 }\right) $.