The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&2&3&-12&1\\& & -10& 35& \color{black}{-115} \\ \hline &\color{blue}{2}&\color{blue}{-7}&\color{blue}{23}&\color{orangered}{-114} \end{array} $$The solution is:
$$ \frac{ 2x^{3}+3x^{2}-12x+1 }{ x+5 } = \color{blue}{2x^{2}-7x+23} \color{red}{~-~} \frac{ \color{red}{ 114 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&3&-12&1\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 2 }&3&-12&1\\& & & & \\ \hline &\color{orangered}{2}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&3&-12&1\\& & \color{blue}{-10} & & \\ \hline &\color{blue}{2}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrr}-5&2&\color{orangered}{ 3 }&-12&1\\& & \color{orangered}{-10} & & \\ \hline &2&\color{orangered}{-7}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ 35 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&3&-12&1\\& & -10& \color{blue}{35} & \\ \hline &2&\color{blue}{-7}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -12 } + \color{orangered}{ 35 } = \color{orangered}{ 23 } $
$$ \begin{array}{c|rrrr}-5&2&3&\color{orangered}{ -12 }&1\\& & -10& \color{orangered}{35} & \\ \hline &2&-7&\color{orangered}{23}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 23 } = \color{blue}{ -115 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&2&3&-12&1\\& & -10& 35& \color{blue}{-115} \\ \hline &2&-7&\color{blue}{23}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 1 } + \color{orangered}{ \left( -115 \right) } = \color{orangered}{ -114 } $
$$ \begin{array}{c|rrrr}-5&2&3&-12&\color{orangered}{ 1 }\\& & -10& 35& \color{orangered}{-115} \\ \hline &\color{blue}{2}&\color{blue}{-7}&\color{blue}{23}&\color{orangered}{-114} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{2}-7x+23 } $ with a remainder of $ \color{red}{ -114 } $.