Divide using synthetic division $$ ( x^{4}-23x^{2}+8x-11 ) \div ( x+5 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&0&-23&8&-11\\& & -5& 25& -10& \color{black}{10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-23x^{2}+8x-11 }{ x+5 } = \color{blue}{x^{3}-5x^{2}+2x-2} \color{red}{~-~} \dfrac{ \color{red}{ 1 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-23&8&-11\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&0&-23&8&-11\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-23&8&-11\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ 0 }&-23&8&-11\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-23&8&-11\\& & -5& \color{blue}{25} & & \\ \hline &1&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -23 } + \color{orangered}{ 25 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-5&1&0&\color{orangered}{ -23 }&8&-11\\& & -5& \color{orangered}{25} & & \\ \hline &1&-5&\color{orangered}{2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 2 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-23&8&-11\\& & -5& 25& \color{blue}{-10} & \\ \hline &1&-5&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-5&1&0&-23&\color{orangered}{ 8 }&-11\\& & -5& 25& \color{orangered}{-10} & \\ \hline &1&-5&2&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-23&8&-11\\& & -5& 25& -10& \color{blue}{10} \\ \hline &1&-5&2&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -11 } + \color{orangered}{ 10 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-5&1&0&-23&8&\color{orangered}{ -11 }\\& & -5& 25& -10& \color{orangered}{10} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-5x^{2}+2x-2 } $ with a remainder of $ \color{red}{ -1 } $.