Divide using synthetic division $$ ( 2x^{4}+17x^{3}+19x^{2}-16x-17 ) \div ( x+7 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-7&2&17&19&-16&-17\\& & -14& -21& 14& \color{black}{14} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{-3} \end{array} $$The solution is:
$$ \frac{ 2x^{4}+17x^{3}+19x^{2}-16x-17 }{ x+7 } = \color{blue}{2x^{3}+3x^{2}-2x-2} \color{red}{~-~} \frac{ \color{red}{ 3 } }{ x+7 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 7 = 0 $ ( $ x = \color{blue}{ -7 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&2&17&19&-16&-17\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-7&\color{orangered}{ 2 }&17&19&-16&-17\\& & & & & \\ \hline &\color{orangered}{2}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 2 } = \color{blue}{ -14 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&2&17&19&-16&-17\\& & \color{blue}{-14} & & & \\ \hline &\color{blue}{2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ \left( -14 \right) } = \color{orangered}{ 3 } $
$$ \begin{array}{c|rrrrr}-7&2&\color{orangered}{ 17 }&19&-16&-17\\& & \color{orangered}{-14} & & & \\ \hline &2&\color{orangered}{3}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ 3 } = \color{blue}{ -21 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&2&17&19&-16&-17\\& & -14& \color{blue}{-21} & & \\ \hline &2&\color{blue}{3}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 19 } + \color{orangered}{ \left( -21 \right) } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-7&2&17&\color{orangered}{ 19 }&-16&-17\\& & -14& \color{orangered}{-21} & & \\ \hline &2&3&\color{orangered}{-2}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&2&17&19&-16&-17\\& & -14& -21& \color{blue}{14} & \\ \hline &2&3&\color{blue}{-2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -16 } + \color{orangered}{ 14 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}-7&2&17&19&\color{orangered}{ -16 }&-17\\& & -14& -21& \color{orangered}{14} & \\ \hline &2&3&-2&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -7 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ 14 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-7}&2&17&19&-16&-17\\& & -14& -21& 14& \color{blue}{14} \\ \hline &2&3&-2&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 14 } = \color{orangered}{ -3 } $
$$ \begin{array}{c|rrrrr}-7&2&17&19&-16&\color{orangered}{ -17 }\\& & -14& -21& 14& \color{orangered}{14} \\ \hline &\color{blue}{2}&\color{blue}{3}&\color{blue}{-2}&\color{blue}{-2}&\color{orangered}{-3} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 2x^{3}+3x^{2}-2x-2 } $ with a remainder of $ \color{red}{ -3 } $.