Divide using synthetic division $$ ( x^{4}-10x^{2}-13x-50 ) \div ( x+5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-5&1&0&-10&-13&-50\\& & -5& 25& -75& \color{black}{440} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{15}&\color{blue}{-88}&\color{orangered}{390} \end{array} $$The solution is:
$$ \frac{ x^{4}-10x^{2}-13x-50 }{ x+5 } = \color{blue}{x^{3}-5x^{2}+15x-88} ~+~ \frac{ \color{red}{ 390 } }{ x+5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-10&-13&-50\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-5&\color{orangered}{ 1 }&0&-10&-13&-50\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-10&-13&-50\\& & \color{blue}{-5} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -5 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrrr}-5&1&\color{orangered}{ 0 }&-10&-13&-50\\& & \color{orangered}{-5} & & & \\ \hline &1&\color{orangered}{-5}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 25 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-10&-13&-50\\& & -5& \color{blue}{25} & & \\ \hline &1&\color{blue}{-5}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 25 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}-5&1&0&\color{orangered}{ -10 }&-13&-50\\& & -5& \color{orangered}{25} & & \\ \hline &1&-5&\color{orangered}{15}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ 15 } = \color{blue}{ -75 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-10&-13&-50\\& & -5& 25& \color{blue}{-75} & \\ \hline &1&-5&\color{blue}{15}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -13 } + \color{orangered}{ \left( -75 \right) } = \color{orangered}{ -88 } $
$$ \begin{array}{c|rrrrr}-5&1&0&-10&\color{orangered}{ -13 }&-50\\& & -5& 25& \color{orangered}{-75} & \\ \hline &1&-5&15&\color{orangered}{-88}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -5 } \cdot \color{blue}{ \left( -88 \right) } = \color{blue}{ 440 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-5}&1&0&-10&-13&-50\\& & -5& 25& -75& \color{blue}{440} \\ \hline &1&-5&15&\color{blue}{-88}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -50 } + \color{orangered}{ 440 } = \color{orangered}{ 390 } $
$$ \begin{array}{c|rrrrr}-5&1&0&-10&-13&\color{orangered}{ -50 }\\& & -5& 25& -75& \color{orangered}{440} \\ \hline &\color{blue}{1}&\color{blue}{-5}&\color{blue}{15}&\color{blue}{-88}&\color{orangered}{390} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-5x^{2}+15x-88 } $ with a remainder of $ \color{red}{ 390 } $.