Divide using synthetic division $$ ( x^{3}-8x^{2}+17x-10 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-5&1&-8&17&-10\\& & -5& 65& \color{black}{-410} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{blue}{82}&\color{orangered}{-420} \end{array} $$The solution is:
$$ \frac{ x^{3}-8x^{2}+17x-10 }{ x+5 } = \color{blue}{x^{2}-13x+82} \color{red}{~-~} \frac{ \color{red}{ 420 } }{ x+5 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 5 = 0 $ ( $ x = \color{blue}{ -5 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-8&17&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-5&\color{orangered}{ 1 }&-8&17&-10\\& & & & \\ \hline &\color{orangered}{1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 1 } = \color{blue}{ -5 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-8&17&-10\\& & \color{blue}{-5} & & \\ \hline &\color{blue}{1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -8 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrr}-5&1&\color{orangered}{ -8 }&17&-10\\& & \color{orangered}{-5} & & \\ \hline &1&\color{orangered}{-13}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 65 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-8&17&-10\\& & -5& \color{blue}{65} & \\ \hline &1&\color{blue}{-13}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 17 } + \color{orangered}{ 65 } = \color{orangered}{ 82 } $
$$ \begin{array}{c|rrrr}-5&1&-8&\color{orangered}{ 17 }&-10\\& & -5& \color{orangered}{65} & \\ \hline &1&-13&\color{orangered}{82}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 82 } = \color{blue}{ -410 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-5}&1&-8&17&-10\\& & -5& 65& \color{blue}{-410} \\ \hline &1&-13&\color{blue}{82}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -410 \right) } = \color{orangered}{ -420 } $
$$ \begin{array}{c|rrrr}-5&1&-8&17&\color{orangered}{ -10 }\\& & -5& 65& \color{orangered}{-410} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{blue}{82}&\color{orangered}{-420} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{2}-13x+82 } $ with a remainder of $ \color{red}{ -420 } $.