Divide using synthetic division $$ ( -x^{4}+18x^{2}-10x+8 ) \div ( x-4 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}4&-1&0&18&-10&8\\& & -4& -16& 8& \color{black}{-8} \\ \hline &\color{blue}{-1}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$The solution is:
$$ \frac{ -x^{4}+18x^{2}-10x+8 }{ x-4 } = \color{blue}{-x^{3}-4x^{2}+2x-2} $$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.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-1&0&18&-10&8\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}4&\color{orangered}{ -1 }&0&18&-10&8\\& & & & & \\ \hline &\color{orangered}{-1}&&&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-1&0&18&-10&8\\& & \color{blue}{-4} & & & \\ \hline &\color{blue}{-1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ \left( -4 \right) } = \color{orangered}{ -4 } $
$$ \begin{array}{c|rrrrr}4&-1&\color{orangered}{ 0 }&18&-10&8\\& & \color{orangered}{-4} & & & \\ \hline &-1&\color{orangered}{-4}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -4 \right) } = \color{blue}{ -16 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-1&0&18&-10&8\\& & -4& \color{blue}{-16} & & \\ \hline &-1&\color{blue}{-4}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 18 } + \color{orangered}{ \left( -16 \right) } = \color{orangered}{ 2 } $
$$ \begin{array}{c|rrrrr}4&-1&0&\color{orangered}{ 18 }&-10&8\\& & -4& \color{orangered}{-16} & & \\ \hline &-1&-4&\color{orangered}{2}&& \end{array} $$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 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-1&0&18&-10&8\\& & -4& -16& \color{blue}{8} & \\ \hline &-1&-4&\color{blue}{2}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 8 } = \color{orangered}{ -2 } $
$$ \begin{array}{c|rrrrr}4&-1&0&18&\color{orangered}{ -10 }&8\\& & -4& -16& \color{orangered}{8} & \\ \hline &-1&-4&2&\color{orangered}{-2}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 4 } \cdot \color{blue}{ \left( -2 \right) } = \color{blue}{ -8 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{4}&-1&0&18&-10&8\\& & -4& -16& 8& \color{blue}{-8} \\ \hline &-1&-4&2&\color{blue}{-2}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 8 } + \color{orangered}{ \left( -8 \right) } = \color{orangered}{ 0 } $
$$ \begin{array}{c|rrrrr}4&-1&0&18&-10&\color{orangered}{ 8 }\\& & -4& -16& 8& \color{orangered}{-8} \\ \hline &\color{blue}{-1}&\color{blue}{-4}&\color{blue}{2}&\color{blue}{-2}&\color{orangered}{0} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -x^{3}-4x^{2}+2x-2 } $ with a remainder of $ \color{red}{ 0 } $.