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