Divide using synthetic division $$ ( 4x^{5}+6x^{4}+5x^{2}-x-10 ) \div ( 2x+3 ) $$
Answer
The synthetic division table is:
$$ \begin{array}{c|rrrrrr}-3&4&6&0&5&-1&-10\\& & -12& 18& -54& 147& \color{black}{-438} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{18}&\color{blue}{-49}&\color{blue}{146}&\color{orangered}{-448} \end{array} $$The solution is:
$$ \frac{ 4x^{5}+6x^{4}+5x^{2}-x-10 }{ x+3 } = \color{blue}{4x^{4}-6x^{3}+18x^{2}-49x+146} \color{red}{~-~} \frac{ \color{red}{ 448 } }{ x+3 } $$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.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & & & & & \\ \hline &&&&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrrrr}-3&\color{orangered}{ 4 }&6&0&5&-1&-10\\& & & & & & \\ \hline &\color{orangered}{4}&&&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 4 } = \color{blue}{ -12 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & \color{blue}{-12} & & & & \\ \hline &\color{blue}{4}&&&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -12 \right) } = \color{orangered}{ -6 } $
$$ \begin{array}{c|rrrrrr}-3&4&\color{orangered}{ 6 }&0&5&-1&-10\\& & \color{orangered}{-12} & & & & \\ \hline &4&\color{orangered}{-6}&&&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -6 \right) } = \color{blue}{ 18 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & -12& \color{blue}{18} & & & \\ \hline &4&\color{blue}{-6}&&&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ 0 } + \color{orangered}{ 18 } = \color{orangered}{ 18 } $
$$ \begin{array}{c|rrrrrr}-3&4&6&\color{orangered}{ 0 }&5&-1&-10\\& & -12& \color{orangered}{18} & & & \\ \hline &4&-6&\color{orangered}{18}&&& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 18 } = \color{blue}{ -54 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & -12& 18& \color{blue}{-54} & & \\ \hline &4&-6&\color{blue}{18}&&& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ 5 } + \color{orangered}{ \left( -54 \right) } = \color{orangered}{ -49 } $
$$ \begin{array}{c|rrrrrr}-3&4&6&0&\color{orangered}{ 5 }&-1&-10\\& & -12& 18& \color{orangered}{-54} & & \\ \hline &4&-6&18&\color{orangered}{-49}&& \end{array} $$Step 8 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ \left( -49 \right) } = \color{blue}{ 147 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & -12& 18& -54& \color{blue}{147} & \\ \hline &4&-6&18&\color{blue}{-49}&& \end{array} $$Step 9 : Add down last column: $ \color{orangered}{ -1 } + \color{orangered}{ 147 } = \color{orangered}{ 146 } $
$$ \begin{array}{c|rrrrrr}-3&4&6&0&5&\color{orangered}{ -1 }&-10\\& & -12& 18& -54& \color{orangered}{147} & \\ \hline &4&-6&18&-49&\color{orangered}{146}& \end{array} $$Step 10 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -3 } \cdot \color{blue}{ 146 } = \color{blue}{ -438 } $.
$$ \begin{array}{c|rrrrrr}\color{blue}{-3}&4&6&0&5&-1&-10\\& & -12& 18& -54& 147& \color{blue}{-438} \\ \hline &4&-6&18&-49&\color{blue}{146}& \end{array} $$Step 11 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ \left( -438 \right) } = \color{orangered}{ -448 } $
$$ \begin{array}{c|rrrrrr}-3&4&6&0&5&-1&\color{orangered}{ -10 }\\& & -12& 18& -54& 147& \color{orangered}{-438} \\ \hline &\color{blue}{4}&\color{blue}{-6}&\color{blue}{18}&\color{blue}{-49}&\color{blue}{146}&\color{orangered}{-448} \end{array} $$Bottom line represents the quotient $ \color{blue}{ 4x^{4}-6x^{3}+18x^{2}-49x+146 } $ with a remainder of $ \color{red}{ -448 } $.