Divide using synthetic division $$ ( -2x^{4}+3x^{3}+4x^{2}+2 ) \div ( x-5 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}5&-2&3&4&0&2\\& & -10& -35& -155& \color{black}{-775} \\ \hline &\color{blue}{-2}&\color{blue}{-7}&\color{blue}{-31}&\color{blue}{-155}&\color{orangered}{-773} \end{array} $$The solution is:
$$ \frac{ -2x^{4}+3x^{3}+4x^{2}+2 }{ x-5 } = \color{blue}{-2x^{3}-7x^{2}-31x-155} \color{red}{~-~} \frac{ \color{red}{ 773 } }{ x-5 } $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-2&3&4&0&2\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}5&\color{orangered}{ -2 }&3&4&0&2\\& & & & & \\ \hline &\color{orangered}{-2}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-2&3&4&0&2\\& & \color{blue}{-10} & & & \\ \hline &\color{blue}{-2}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -10 \right) } = \color{orangered}{ -7 } $
$$ \begin{array}{c|rrrrr}5&-2&\color{orangered}{ 3 }&4&0&2\\& & \color{orangered}{-10} & & & \\ \hline &-2&\color{orangered}{-7}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -7 \right) } = \color{blue}{ -35 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-2&3&4&0&2\\& & -10& \color{blue}{-35} & & \\ \hline &-2&\color{blue}{-7}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ \left( -35 \right) } = \color{orangered}{ -31 } $
$$ \begin{array}{c|rrrrr}5&-2&3&\color{orangered}{ 4 }&0&2\\& & -10& \color{orangered}{-35} & & \\ \hline &-2&-7&\color{orangered}{-31}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -31 \right) } = \color{blue}{ -155 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-2&3&4&0&2\\& & -10& -35& \color{blue}{-155} & \\ \hline &-2&-7&\color{blue}{-31}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -155 \right) } = \color{orangered}{ -155 } $
$$ \begin{array}{c|rrrrr}5&-2&3&4&\color{orangered}{ 0 }&2\\& & -10& -35& \color{orangered}{-155} & \\ \hline &-2&-7&-31&\color{orangered}{-155}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 5 } \cdot \color{blue}{ \left( -155 \right) } = \color{blue}{ -775 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{5}&-2&3&4&0&2\\& & -10& -35& -155& \color{blue}{-775} \\ \hline &-2&-7&-31&\color{blue}{-155}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 2 } + \color{orangered}{ \left( -775 \right) } = \color{orangered}{ -773 } $
$$ \begin{array}{c|rrrrr}5&-2&3&4&0&\color{orangered}{ 2 }\\& & -10& -35& -155& \color{orangered}{-775} \\ \hline &\color{blue}{-2}&\color{blue}{-7}&\color{blue}{-31}&\color{blue}{-155}&\color{orangered}{-773} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -2x^{3}-7x^{2}-31x-155 } $ with a remainder of $ \color{red}{ -773 } $.