Divide using synthetic division $$ ( -x^{3}+11x-20 ) \div ( x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}-4&-1&0&11&-20\\& & 4& -16& \color{black}{20} \\ \hline &\color{blue}{-1}&\color{blue}{4}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{3}+11x-20 }{ x+4 } = \color{blue}{-x^{2}+4x-5} $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 4 = 0 $ ( $ x = \color{blue}{ -4 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-1&0&11&-20\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-4&\color{orangered}{ -1 }&0&11&-20\\& & & & \\ \hline &\color{orangered}{-1}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-1&0&11&-20\\& & \color{blue}{4} & & \\ \hline &\color{blue}{-1}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 4 } = \color{orangered}{ 4 } $
$$ \begin{array}{c|rrrr}-4&-1&\color{orangered}{ 0 }&11&-20\\& & \color{orangered}{4} & & \\ \hline &-1&\color{orangered}{4}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 4 } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-1&0&11&-20\\& & 4& \color{blue}{-16} & \\ \hline &-1&\color{blue}{4}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 11 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ -5 } $
$$ \begin{array}{c|rrrr}-4&-1&0&\color{orangered}{ 11 }&-20\\& & 4& \color{orangered}{-16} & \\ \hline &-1&4&\color{orangered}{-5}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -5 \right) } = \color{blue}{ 20 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-4}&-1&0&11&-20\\& & 4& -16& \color{blue}{20} \\ \hline &-1&4&\color{blue}{-5}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -20 } + \color{orangered}{ 20 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrr}-4&-1&0&11&\color{orangered}{ -20 }\\& & 4& -16& \color{orangered}{20} \\ \hline &\color{blue}{-1}&\color{blue}{4}&\color{blue}{-5}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{2}+4x-5 } $ with a remainder of $ \color{red}{ 0 } $.