Divide using synthetic division $$ ( 5x^{4}-3x^{3}-17x^{2}+23x+10 ) \div ( x+2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-2&5&-3&-17&23&10\\& & -10& 26& -18& \color{black}{-10} \\ \hline &\color{blue}{5}&\color{blue}{-13}&\color{blue}{9}&\color{blue}{5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ 5x^{4}-3x^{3}-17x^{2}+23x+10 }{ x+2 } = \color{blue}{5x^{3}-13x^{2}+9x+5} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&5&-3&-17&23&10\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-2&\color{orangered}{ 5 }&-3&-17&23&10\\& & & & & \\ \hline &\color{orangered}{5}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&5&-3&-17&23&10\\& & \color{blue}{-10} & & & \\ \hline &\color{blue}{5}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -3 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrrr}-2&5&\color{orangered}{ -3 }&-17&23&10\\& & \color{orangered}{-10} & & & \\ \hline &5&\color{orangered}{-13}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 26 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&5&-3&-17&23&10\\& & -10& \color{blue}{26} & & \\ \hline &5&\color{blue}{-13}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -17 } + \color{orangered}{ 26 } = \color{orangered}{ 9 } $
$$ \begin{array}{c|rrrrr}-2&5&-3&\color{orangered}{ -17 }&23&10\\& & -10& \color{orangered}{26} & & \\ \hline &5&-13&\color{orangered}{9}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 9 } = \color{blue}{ -18 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&5&-3&-17&23&10\\& & -10& 26& \color{blue}{-18} & \\ \hline &5&-13&\color{blue}{9}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 23 } + \color{orangered}{ \left( -18 \right) } = \color{orangered}{ 5 } $
$$ \begin{array}{c|rrrrr}-2&5&-3&-17&\color{orangered}{ 23 }&10\\& & -10& 26& \color{orangered}{-18} & \\ \hline &5&-13&9&\color{orangered}{5}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -2 } \cdot \color{blue}{ 5 } = \color{blue}{ -10 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-2}&5&-3&-17&23&10\\& & -10& 26& -18& \color{blue}{-10} \\ \hline &5&-13&9&\color{blue}{5}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 10 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-2&5&-3&-17&23&\color{orangered}{ 10 }\\& & -10& 26& -18& \color{orangered}{-10} \\ \hline &\color{blue}{5}&\color{blue}{-13}&\color{blue}{9}&\color{blue}{5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 5x^{3}-13x^{2}+9x+5 } $ with a remainder of $ \color{red}{ 0 } $.