Divide using synthetic division $$ ( -x^{2}+2x+3 ) \div ( -x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrr}-4&-1&2&3\\& & 4& \color{black}{-24} \\ \hline &\color{blue}{-1}&\color{blue}{6}&\color{orangered}{-21} \end{array} $$The solution is:
$$ \frac{ -x^{2}+2x+3 }{ x+4 } = \color{blue}{-x+6} \color{red}{~-~} \frac{ \color{red}{ 21 } }{ x+4 } $$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.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-1&2&3\\& & & \\ \hline &&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrr}-4&\color{orangered}{ -1 }&2&3\\& & & \\ \hline &\color{orangered}{-1}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-1&2&3\\& & \color{blue}{4} & \\ \hline &\color{blue}{-1}&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ 4 } = \color{orangered}{ 6 } $
$$ \begin{array}{c|rrr}-4&-1&\color{orangered}{ 2 }&3\\& & \color{orangered}{4} & \\ \hline &-1&\color{orangered}{6}& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ 6 } = \color{blue}{ -24 } $.
$$ \begin{array}{c|rrr}\color{blue}{-4}&-1&2&3\\& & 4& \color{blue}{-24} \\ \hline &-1&\color{blue}{6}& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -24 \right) } = \color{orangered}{ -21 } $
$$ \begin{array}{c|rrr}-4&-1&2&\color{orangered}{ 3 }\\& & 4& \color{orangered}{-24} \\ \hline &\color{blue}{-1}&\color{blue}{6}&\color{orangered}{-21} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x+6 } $ with a remainder of $ \color{red}{ -21 } $.