Divide using synthetic division $$ ( -3x^{4}-10x^{3}+7x^{2}-4x ) \div ( -x+4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-4&-3&-10&7&-4&0\\& & 12& -8& 4& \color{black}{0} \\ \hline &\color{blue}{-3}&\color{blue}{2}&\color{blue}{-1}&\color{blue}{0}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -3x^{4}-10x^{3}+7x^{2}-4x }{ x+4 } = \color{blue}{-3x^{3}+2x^{2}-x} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-3&-10&7&-4&0\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-4&\color{orangered}{ -3 }&-10&7&-4&0\\& & & & & \\ \hline &\color{orangered}{-3}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -3 \right) } = \color{blue}{ 12 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-3&-10&7&-4&0\\& & \color{blue}{12} & & & \\ \hline &\color{blue}{-3}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 12 } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}-4&-3&\color{orangered}{ -10 }&7&-4&0\\& & \color{orangered}{12} & & & \\ \hline &-3&\color{orangered}{2}&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-3&-10&7&-4&0\\& & 12& \color{blue}{-8} & & \\ \hline &-3&\color{blue}{2}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 7 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrrr}-4&-3&-10&\color{orangered}{ 7 }&-4&0\\& & 12& \color{orangered}{-8} & & \\ \hline &-3&2&\color{orangered}{-1}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -4 } \cdot \color{blue}{ \left( -1 \right) } = \color{blue}{ 4 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-3&-10&7&-4&0\\& & 12& -8& \color{blue}{4} & \\ \hline &-3&2&\color{blue}{-1}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -4 } + \color{orangered}{ 4 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-4&-3&-10&7&\color{orangered}{ -4 }&0\\& & 12& -8& \color{orangered}{4} & \\ \hline &-3&2&-1&\color{orangered}{0}& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-4}&-3&-10&7&-4&0\\& & 12& -8& 4& \color{blue}{0} \\ \hline &-3&2&-1&\color{blue}{0}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 0 } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}-4&-3&-10&7&-4&\color{orangered}{ 0 }\\& & 12& -8& 4& \color{orangered}{0} \\ \hline &\color{blue}{-3}&\color{blue}{2}&\color{blue}{-1}&\color{blue}{0}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -3x^{3}+2x^{2}-x } $ with a remainder of $ \color{red}{ 0 } $.