Divide using synthetic division $$ ( 2x^{6}+5x-5 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrrr}-2&2&0&0&0&0&5&-5\\& & -4& 8& -16& 32& -64& \color{black}{118} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{8}&\color{blue}{-16}&\color{blue}{32}&\color{blue}{-59}&\color{orangered}{113} \end{array} $$The solution is:
$$ \frac{ 2x^{6}+5x-5 }{ x+2 } = \color{blue}{2x^{5}-4x^{4}+8x^{3}-16x^{2}+32x-59} ~+~ \frac{ \color{red}{ 113 } }{ x+2 } $$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.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & & & & & & \\ \hline &&&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrrr}-2&\color{orangered}{ 2 }&0&0&0&0&5&-5\\& & & & & & & \\ \hline &\color{orangered}{2}&&&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & \color{blue}{-4} & & & & & \\ \hline &\color{blue}{2}&&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrrrr}-2&2&\color{orangered}{ 0 }&0&0&0&5&-5\\& & \color{orangered}{-4} & & & & & \\ \hline &2&\color{orangered}{-4}&&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ 8 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & -4& \color{blue}{8} & & & & \\ \hline &2&\color{blue}{-4}&&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 8 } = \color{orangered}{ 8 } $
$$ \begin{array}{c|rrrrrrr}-2&2&0&\color{orangered}{ 0 }&0&0&5&-5\\& & -4& \color{orangered}{8} & & & & \\ \hline &2&-4&\color{orangered}{8}&&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 8 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & -4& 8& \color{blue}{-16} & & & \\ \hline &2&-4&\color{blue}{8}&&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -16 } $
$$ \begin{array}{c|rrrrrrr}-2&2&0&0&\color{orangered}{ 0 }&0&5&-5\\& & -4& 8& \color{orangered}{-16} & & & \\ \hline &2&-4&8&\color{orangered}{-16}&&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -16 \right) } = \color{blue}{ 32 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & -4& 8& -16& \color{blue}{32} & & \\ \hline &2&-4&8&\color{blue}{-16}&&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 32 } = \color{orangered}{ 32 } $
$$ \begin{array}{c|rrrrrrr}-2&2&0&0&0&\color{orangered}{ 0 }&5&-5\\& & -4& 8& -16& \color{orangered}{32} & & \\ \hline &2&-4&8&-16&\color{orangered}{32}&& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 32 } = \color{blue}{ -64 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & -4& 8& -16& 32& \color{blue}{-64} & \\ \hline &2&-4&8&-16&\color{blue}{32}&& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -64 \right) } = \color{orangered}{ -59 } $
$$ \begin{array}{c|rrrrrrr}-2&2&0&0&0&0&\color{orangered}{ 5 }&-5\\& & -4& 8& -16& 32& \color{orangered}{-64} & \\ \hline &2&-4&8&-16&32&\color{orangered}{-59}& \end{array} $$Step 12 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -59 \right) } = \color{blue}{ 118 } $.
$$ \begin{array}{c|rrrrrrr}\color{blue}{-2}&2&0&0&0&0&5&-5\\& & -4& 8& -16& 32& -64& \color{blue}{118} \\ \hline &2&-4&8&-16&32&\color{blue}{-59}& \end{array} $$Step 13 : Add down last column: $ \color{orangered}{ -5 } + \color{orangered}{ 118 } = \color{orangered}{ 113 } $
$$ \begin{array}{c|rrrrrrr}-2&2&0&0&0&0&5&\color{orangered}{ -5 }\\& & -4& 8& -16& 32& -64& \color{orangered}{118} \\ \hline &\color{blue}{2}&\color{blue}{-4}&\color{blue}{8}&\color{blue}{-16}&\color{blue}{32}&\color{blue}{-59}&\color{orangered}{113} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{5}-4x^{4}+8x^{3}-16x^{2}+32x-59 } $ with a remainder of $ \color{red}{ 113 } $.