Divide using synthetic division $$ ( x^{4}-10x^{3}+42x^{2}-82x+65 ) \div ( -2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrr}-3&1&-10&42&-82&65\\& & -3& 39& -243& \color{black}{975} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{blue}{81}&\color{blue}{-325}&\color{orangered}{1040} \end{array} $$The solution is:
$$ \frac{ x^{4}-10x^{3}+42x^{2}-82x+65 }{ x+3 } = \color{blue}{x^{3}-13x^{2}+81x-325} ~+~ \frac{ \color{red}{ 1040 } }{ x+3 } $$Explanation
Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 3 = 0 $ ( $ x = \color{blue}{ -3 } $ ) at the left.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-10&42&-82&65\\& & & & & \\ \hline &&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrr}-3&\color{orangered}{ 1 }&-10&42&-82&65\\& & & & & \\ \hline &\color{orangered}{1}&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 1 } = \color{blue}{ -3 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-10&42&-82&65\\& & \color{blue}{-3} & & & \\ \hline &\color{blue}{1}&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -3 \right) } = \color{orangered}{ -13 } $
$$ \begin{array}{c|rrrrr}-3&1&\color{orangered}{ -10 }&42&-82&65\\& & \color{orangered}{-3} & & & \\ \hline &1&\color{orangered}{-13}&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -13 \right) } = \color{blue}{ 39 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-10&42&-82&65\\& & -3& \color{blue}{39} & & \\ \hline &1&\color{blue}{-13}&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 42 } + \color{orangered}{ 39 } = \color{orangered}{ 81 } $
$$ \begin{array}{c|rrrrr}-3&1&-10&\color{orangered}{ 42 }&-82&65\\& & -3& \color{orangered}{39} & & \\ \hline &1&-13&\color{orangered}{81}&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 81 } = \color{blue}{ -243 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-10&42&-82&65\\& & -3& 39& \color{blue}{-243} & \\ \hline &1&-13&\color{blue}{81}&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -82 } + \color{orangered}{ \left( -243 \right) } = \color{orangered}{ -325 } $
$$ \begin{array}{c|rrrrr}-3&1&-10&42&\color{orangered}{ -82 }&65\\& & -3& 39& \color{orangered}{-243} & \\ \hline &1&-13&81&\color{orangered}{-325}& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -325 \right) } = \color{blue}{ 975 } $.
$$ \begin{array}{c|rrrrr}\color{blue}{-3}&1&-10&42&-82&65\\& & -3& 39& -243& \color{blue}{975} \\ \hline &1&-13&81&\color{blue}{-325}& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ 65 } + \color{orangered}{ 975 } = \color{orangered}{ 1040 } $
$$ \begin{array}{c|rrrrr}-3&1&-10&42&-82&\color{orangered}{ 65 }\\& & -3& 39& -243& \color{orangered}{975} \\ \hline &\color{blue}{1}&\color{blue}{-13}&\color{blue}{81}&\color{blue}{-325}&\color{orangered}{1040} \end{array} $$Bottom line represents the quotient $ \color{blue}{ x^{3}-13x^{2}+81x-325 } $ with a remainder of $ \color{red}{ 1040 } $.