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