Divide using synthetic division $$ ( -9x^{3}+16x^{2}+3x-10 ) \div ( x-9 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrr}9&-9&16&3&-10\\& & -81& -585& \color{black}{-5238} \\ \hline &\color{blue}{-9}&\color{blue}{-65}&\color{blue}{-582}&\color{orangered}{-5248} \end{array} $$The solution is:
$$ \frac{ -9x^{3}+16x^{2}+3x-10 }{ x-9 } = \color{blue}{-9x^{2}-65x-582} \color{red}{~-~} \frac{ \color{red}{ 5248 } }{ x-9 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x -9 = 0 $ ( $ x = \color{blue}{ 9 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{9}&-9&16&3&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}9&\color{orangered}{ -9 }&16&3&-10\\& & & & \\ \hline &\color{orangered}{-9}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ -81 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&-9&16&3&-10\\& & \color{blue}{-81} & & \\ \hline &\color{blue}{-9}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 16 } + \color{orangered}{ \left( -81 \right) } = \color{orangered}{ -65 } $
$$ \begin{array}{c|rrrr}9&-9&\color{orangered}{ 16 }&3&-10\\& & \color{orangered}{-81} & & \\ \hline &-9&\color{orangered}{-65}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -65 \right) } = \color{blue}{ -585 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&-9&16&3&-10\\& & -81& \color{blue}{-585} & \\ \hline &-9&\color{blue}{-65}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 3 } + \color{orangered}{ \left( -585 \right) } = \color{orangered}{ -582 } $
$$ \begin{array}{c|rrrr}9&-9&16&\color{orangered}{ 3 }&-10\\& & -81& \color{orangered}{-585} & \\ \hline &-9&-65&\color{orangered}{-582}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ 9 } \cdot \color{blue}{ \left( -582 \right) } = \color{blue}{ -5238 } $.
$$ \begin{array}{c|rrrr}\color{blue}{9}&-9&16&3&-10\\& & -81& -585& \color{blue}{-5238} \\ \hline &-9&-65&\color{blue}{-582}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -5238 \right) } = \color{orangered}{ -5248 } $
$$ \begin{array}{c|rrrr}9&-9&16&3&\color{orangered}{ -10 }\\& & -81& -585& \color{orangered}{-5238} \\ \hline &\color{blue}{-9}&\color{blue}{-65}&\color{blue}{-582}&\color{orangered}{-5248} \end{array} $$Bottom line represents the quotient $ \color{blue}{ -9x^{2}-65x-582 } $ with a remainder of $ \color{red}{ -5248 } $.