Divide using synthetic division $$ ( 10x^{4}+5x^{3}+4x^{2}-9 ) \div ( x-1 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}1&10&5&4&0&-9\\& & 10& 15& 19& \color{black}{19} \\ \hline &\color{blue}{10}&\color{blue}{15}&\color{blue}{19}&\color{blue}{19}&\color{orangered}{10} \end{array} $$The solution is:
$$ \frac{ 10x^{4}+5x^{3}+4x^{2}-9 }{ x-1 } = \color{blue}{10x^{3}+15x^{2}+19x+19} ~+~ \frac{ \color{red}{ 10 } }{ x-1 } $$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}{ 10 } = \color{orangered}{ 15 } $
$$ \begin{array}{c|rrrrr}1&10&\color{orangered}{ 5 }&4&0&-9\\& & \color{orangered}{10} & & & \\ \hline &10&\color{orangered}{15}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 15 } = \color{blue}{ 15 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&10&5&4&0&-9\\& & 10& \color{blue}{15} & & \\ \hline &10&\color{blue}{15}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 4 } + \color{orangered}{ 15 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}1&10&5&\color{orangered}{ 4 }&0&-9\\& & 10& \color{orangered}{15} & & \\ \hline &10&15&\color{orangered}{19}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&10&5&4&0&-9\\& & 10& 15& \color{blue}{19} & \\ \hline &10&15&\color{blue}{19}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 19 } = \color{orangered}{ 19 } $
$$ \begin{array}{c|rrrrr}1&10&5&4&\color{orangered}{ 0 }&-9\\& & 10& 15& \color{orangered}{19} & \\ \hline &10&15&19&\color{orangered}{19}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 1 } \cdot \color{blue}{ 19 } = \color{blue}{ 19 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{1}&10&5&4&0&-9\\& & 10& 15& 19& \color{blue}{19} \\ \hline &10&15&19&\color{blue}{19}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -9 } + \color{orangered}{ 19 } = \color{orangered}{ 10 } $
$$ \begin{array}{c|rrrrr}1&10&5&4&0&\color{orangered}{ -9 }\\& & 10& 15& 19& \color{orangered}{19} \\ \hline &\color{blue}{10}&\color{blue}{15}&\color{blue}{19}&\color{blue}{19}&\color{orangered}{10} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 10x^{3}+15x^{2}+19x+19 } $ with a remainder of $ \color{red}{ 10 } $.