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