Divide using synthetic division $$ ( 9x^{3}+90x^{2}-2 ) \div ( x-2 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}2&9&90&0&-2\\& & 18& 216& \color{black}{432} \\ \hline &\color{blue}{9}&\color{blue}{108}&\color{blue}{216}&\color{orangered}{430} \end{array} $$The solution is:
$$ \frac{ 9x^{3}+90x^{2}-2 }{ x-2 } = \color{blue}{9x^{2}+108x+216} ~+~ \frac{ \color{red}{ 430 } }{ x-2 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -2 = 0 $ ( $ x = \color{blue}{ 2 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{2}&9&90&0&-2\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}2&\color{orangered}{ 9 }&90&0&-2\\& & & & \\ \hline &\color{orangered}{9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 9 } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&9&90&0&-2\\& & \color{blue}{18} & & \\ \hline &\color{blue}{9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 90 } + \color{orangered}{ 18 } = \color{orangered}{ 108 } $
$$ \begin{array}{c|rrrr}2&9&\color{orangered}{ 90 }&0&-2\\& & \color{orangered}{18} & & \\ \hline &9&\color{orangered}{108}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 108 } = \color{blue}{ 216 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&9&90&0&-2\\& & 18& \color{blue}{216} & \\ \hline &9&\color{blue}{108}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 216 } = \color{orangered}{ 216 } $
$$ \begin{array}{c|rrrr}2&9&90&\color{orangered}{ 0 }&-2\\& & 18& \color{orangered}{216} & \\ \hline &9&108&\color{orangered}{216}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 2 } \cdot \color{blue}{ 216 } = \color{blue}{ 432 } $.
$$ \begin{array}{c|rrrr}\color{blue}{2}&9&90&0&-2\\& & 18& 216& \color{blue}{432} \\ \hline &9&108&\color{blue}{216}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -2 } + \color{orangered}{ 432 } = \color{orangered}{ 430 } $
$$ \begin{array}{c|rrrr}2&9&90&0&\color{orangered}{ -2 }\\& & 18& 216& \color{orangered}{432} \\ \hline &\color{blue}{9}&\color{blue}{108}&\color{blue}{216}&\color{orangered}{430} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 9x^{2}+108x+216 } $ with a remainder of $ \color{red}{ 430 } $.