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