Divide using synthetic division $$ ( -x^{3}+75x-250 ) \div ( x+10 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-10&-1&0&75&-250\\& & 10& -100& \color{black}{250} \\ \hline &\color{blue}{-1}&\color{blue}{10}&\color{blue}{-25}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}+75x-250 }{ x+10 } = \color{blue}{-x^{2}+10x-25} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 10 = 0 $ ( $ x = \color{blue}{ -10 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&-1&0&75&-250\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-10&\color{orangered}{ -1 }&0&75&-250\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 10 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&-1&0&75&-250\\& & \color{blue}{10} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 10 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrr}-10&-1&\color{orangered}{ 0 }&75&-250\\& & \color{orangered}{10} & & \\ \hline &-1&\color{orangered}{10}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ 10 } = \color{blue}{ -100 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&-1&0&75&-250\\& & 10& \color{blue}{-100} & \\ \hline &-1&\color{blue}{10}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 75 } + \color{orangered}{ \left( -100 \right) } = \color{orangered}{ -25 } $
$$ \begin{array}{c|rrrr}-10&-1&0&\color{orangered}{ 75 }&-250\\& & 10& \color{orangered}{-100} & \\ \hline &-1&10&\color{orangered}{-25}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -10 } \cdot \color{blue}{ \left( -25 \right) } = \color{blue}{ 250 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-10}&-1&0&75&-250\\& & 10& -100& \color{blue}{250} \\ \hline &-1&10&\color{blue}{-25}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -250 } + \color{orangered}{ 250 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-10&-1&0&75&\color{orangered}{ -250 }\\& & 10& -100& \color{orangered}{250} \\ \hline &\color{blue}{-1}&\color{blue}{10}&\color{blue}{-25}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+10x-25 } $ with a remainder of $ \color{red}{ 0 } $.