Divide using synthetic division $$ ( 2x^{4}-13x^{3}-13x^{2}-13x-42 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&2&-13&-13&-13&-42\\& & -4& 34& -42& \color{black}{110} \\ \hline &\color{blue}{2}&\color{blue}{-17}&\color{blue}{21}&\color{blue}{-55}&\color{orangered}{68} \end{array} $$The solution is:
$$ \frac{ 2x^{4}-13x^{3}-13x^{2}-13x-42 }{ x+2 } = \color{blue}{2x^{3}-17x^{2}+21x-55} ~+~ \frac{ \color{red}{ 68 } }{ x+2 } $$Explanation
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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-13&-13&-13&-42\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 2 }&-13&-13&-13&-42\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-13&-13&-13&-42\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -17 } $
$$ \begin{array}{c|rrrrr}-2&2&\color{orangered}{ -13 }&-13&-13&-42\\& & \color{orangered}{-4} & & & \\ \hline &2&\color{orangered}{-17}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -17 \right) } = \color{blue}{ 34 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-13&-13&-13&-42\\& & -4& \color{blue}{34} & & \\ \hline &2&\color{blue}{-17}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ 34 } = \color{orangered}{ 21 } $
$$ \begin{array}{c|rrrrr}-2&2&-13&\color{orangered}{ -13 }&-13&-42\\& & -4& \color{orangered}{34} & & \\ \hline &2&-17&\color{orangered}{21}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 21 } = \color{blue}{ -42 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-13&-13&-13&-42\\& & -4& 34& \color{blue}{-42} & \\ \hline &2&-17&\color{blue}{21}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -42 \right) } = \color{orangered}{ -55 } $
$$ \begin{array}{c|rrrrr}-2&2&-13&-13&\color{orangered}{ -13 }&-42\\& & -4& 34& \color{orangered}{-42} & \\ \hline &2&-17&21&\color{orangered}{-55}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -55 \right) } = \color{blue}{ 110 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&2&-13&-13&-13&-42\\& & -4& 34& -42& \color{blue}{110} \\ \hline &2&-17&21&\color{blue}{-55}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -42 } + \color{orangered}{ 110 } = \color{orangered}{ 68 } $
$$ \begin{array}{c|rrrrr}-2&2&-13&-13&-13&\color{orangered}{ -42 }\\& & -4& 34& -42& \color{orangered}{110} \\ \hline &\color{blue}{2}&\color{blue}{-17}&\color{blue}{21}&\color{blue}{-55}&\color{orangered}{68} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}-17x^{2}+21x-55 } $ with a remainder of $ \color{red}{ 68 } $.