Divide using synthetic division $$ ( x^{4}-6x^{3}+3x^{2}+26x-24 ) \div ( x+2 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&1&-6&3&26&-24\\& & -2& 16& -38& \color{black}{24} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{19}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$The solution is:
$$ \dfrac{ x^{4}-6x^{3}+3x^{2}+26x-24 }{ x+2 } = \color{blue}{x^{3}-8x^{2}+19x-12} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&3&26&-24\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 1 }&-6&3&26&-24\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 1 } = \color{blue}{ -2 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&3&26&-24\\& & \color{blue}{-2} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -6 } + \color{orangered}{ \left( -2 \right) } = \color{orangered}{ -8 } $
$$ \begin{array}{c|rrrrr}-2&1&\color{orangered}{ -6 }&3&26&-24\\& & \color{orangered}{-2} & & & \\ \hline &1&\color{orangered}{-8}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&3&26&-24\\& & -2& \color{blue}{16} & & \\ \hline &1&\color{blue}{-8}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ 16 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&\color{orangered}{ 3 }&26&-24\\& & -2& \color{orangered}{16} & & \\ \hline &1&-8&\color{orangered}{19}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 19 } = \color{blue}{ -38 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&3&26&-24\\& & -2& 16& \color{blue}{-38} & \\ \hline &1&-8&\color{blue}{19}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 26 } + \color{orangered}{ \left( -38 \right) } = \color{orangered}{ -12 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&3&\color{orangered}{ 26 }&-24\\& & -2& 16& \color{orangered}{-38} & \\ \hline &1&-8&19&\color{orangered}{-12}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -12 \right) } = \color{blue}{ 24 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&1&-6&3&26&-24\\& & -2& 16& -38& \color{blue}{24} \\ \hline &1&-8&19&\color{blue}{-12}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -24 } + \color{orangered}{ 24 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&1&-6&3&26&\color{orangered}{ -24 }\\& & -2& 16& -38& \color{orangered}{24} \\ \hline &\color{blue}{1}&\color{blue}{-8}&\color{blue}{19}&\color{blue}{-12}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-8x^{2}+19x-12 } $ with a remainder of $ \color{red}{ 0 } $.