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